EFFICIENT NEARFIELD WIRELESS ENERGY TRANSFER USING ADIABATIC SYSTEM VARIATIONS

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First Claim
1. A method for transferring energy wirelessly, the method comprising:
 transferring energy wirelessly from a first resonator structure to an intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ_{1B};
transferring energy wirelessly from the intermediate resonator structure to a second resonator structure, wherein the coupling rate between the intermediate resonator structure and the second resonator structure is κ_{B2}; and
during the wireless energy transfers, adjusting at least one of the coupling rates κ_{1B }and κ_{B2 }to reduce energy accumulation in the intermediate resonator structure and improve wireless energy transfer from the first resonator structure to the second resonator structure through the intermediate resonator structure.
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Abstract
Disclosed is a method for transferring energy wirelessly including transferring energy wirelessly from a first resonator structure to an intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ1B, transferring energy wirelessly from the intermediate resonator structure to a second resonator structure, wherein the coupling rate between the intermediate resonator structure and the second resonator structure is κB2, and during the wireless energy transfers, adjusting at least one of the coupling rates κ1B and κB2 to reduce energy accumulation in the intermediate resonator structure and improve wireless energy transfer from the first resonator structure to the second resonator structure through the intermediate resonator structure.
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Witricity Corporation

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Massachusetts Institute of Technology

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Witricity Corporation

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Massachusetts Institute of Technology

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Massachusetts Institute of Technology

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Massachusetts Institute of Technology

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Massachusetts Institute of Technology

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Witricity Corporation

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Witricity Corporation

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Honda Motor Company

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

Wireless energy transfer using variable size resonators and systems monitoring  
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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Duracell U.S. Operations Inc.

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Sekisui Chemical Company Limited

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Witricity Corporation

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Witricity Corporation

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General Electric Company

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Massachusetts Institute of Technology

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Massachusetts Institute of Technology

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Massachusetts Institute of Technology

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Witricity Corporation

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Wireless power transfer systems with shaped magnetic components  
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Witricity Corporation

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Witricity Corporation

Wireless energy transfer for wearables  
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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

Wireless power harvesting and transmission with heterogeneous signals  
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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Massachusetts Institute of Technology

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Massachusetts Institute of Technology

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Witricity Corporation

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Massachusetts Institute of Technology

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Massachusetts Institute of Technology

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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Witricity Corporation

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US 10,300,800 B2
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Witricity Corporation

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Witricity Corporation

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40 Claims
 1. A method for transferring energy wirelessly, the method comprising:
 transferring energy wirelessly from a first resonator structure to an intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ_{1B};
transferring energy wirelessly from the intermediate resonator structure to a second resonator structure, wherein the coupling rate between the intermediate resonator structure and the second resonator structure is κ_{B2}; and
during the wireless energy transfers, adjusting at least one of the coupling rates κ_{1B }and κ_{B2 }to reduce energy accumulation in the intermediate resonator structure and improve wireless energy transfer from the first resonator structure to the second resonator structure through the intermediate resonator structure.  View Dependent Claims (2)
 transferring energy wirelessly from a first resonator structure to an intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ_{1B};
 3. The method of claims 1, wherein the adjustment of at least one of the coupling rates κ1B and κB2 maintains energy distribution in the field of the threeresonator system in an eigenstate having substantially no energy in the intermediate resonator structure.
 31. A method for transferring energy wirelessly, the method comprising:
 transferring energy wirelessly from a first resonator structure to a intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ_{1B};
transferring energy wirelessly from the intermediate resonator structure to a second resonator, wherein the coupling rate between the intermediate resonator structure and the second resonator structure is κ_{B2}; and
during the wireless energy transfers, adjusting at least one of the coupling rates κ_{1B }and κ_{B2 }to cause an energy distribution in the field of the threeresonator system to have substantially no energy in the intermediate resonator structure while wirelessly transferring energy from the first resonator structure to the second resonator structure through the intermediate resonator structure.  View Dependent Claims (32, 33, 34, 35)
 transferring energy wirelessly from a first resonator structure to a intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ_{1B};
 36. An apparatus comprising:
 first, intermediate, and second resonator structures, wherein a coupling rate between the first resonator structure and the intermediate resonator structure is κ_{1B }and a coupling rate between the intermediate resonator structure and the second resonator structure is κ_{B2}; and
means for adjusting at least one of the coupling rates κ_{1B }and κ_{B2 }during wireless energy transfers among the resonator structures to cause an energy distribution in the field of the threeresonator system to have substantially no energy in the intermediate resonator structure while wirelessly transferring energy from the first resonator structure to the second resonator structure through the intermediate resonator structure.
 first, intermediate, and second resonator structures, wherein a coupling rate between the first resonator structure and the intermediate resonator structure is κ_{1B }and a coupling rate between the intermediate resonator structure and the second resonator structure is κ_{B2}; and
 37. An apparatus comprising:
 first, intermediate, and second resonator structures, wherein a coupling rate between the first resonator structure and the intermediate resonator structure is κ_{1B }and a coupling rate between the intermediate resonator structure and the second resonator structure is κ_{B2}; and
means for adjusting at least one of the coupling rates κ_{1B }and κ_{B2 }during wireless energy transfers among the resonator structures to reduce energy accumulation in the intermediate resonator structure and improve wireless energy transfer from the first resonator structure to the second resonator structure through the intermediate resonator structure.  View Dependent Claims (38, 39, 40)
 first, intermediate, and second resonator structures, wherein a coupling rate between the first resonator structure and the intermediate resonator structure is κ_{1B }and a coupling rate between the intermediate resonator structure and the second resonator structure is κ_{B2}; and
1 Specification
Pursuant to U.S.C. §119(e), this application claims priority to U.S. Provisional Application Ser. No. 61/101,809, filed Oct. 1, 2008. The contents of the prior application is incorporated herein by reference in its entirety.
BACKGROUNDThe disclosure relates to wireless energy transfer. Wireless energy transfer can for example, be useful in such applications as providing power to autonomous electrical or electronic devices.
Radiative modes of omnidirectional antennas (which work very well for information transfer) are not suitable for such energy transfer, because a vast majority of energy is wasted into free space. Directed radiation modes, using lasers or highlydirectional antennas, can be efficiently used for energy transfer, even for long distances (transfer distance L<sub>TRANS</sub>>>L<sub>DEV</sub>, where L<sub>DEV </sub>is the characteristic size of the device and/or the source), but may require existence of an uninterruptible lineofsight and a complicated tracking system in the case of mobile objects. Some transfer schemes rely on induction, but are typically restricted to very closerange (L<sub>TRANS</sub><<L<sub>DEV</sub>) or low power (˜mW) energy transfers.
The rapid development of autonomous electronics of recent years (e.g. laptops, cellphones, household robots, that all typically rely on chemical energy storage) has led to an increased need for wireless energy transfer.
SUMMARYDisclosed is a method for transferring energy wirelessly. The method includes i) transferring energy wirelessly from a first resonator structure to an intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ<sub>1B</sub>; ii) transferring energy wirelessly from the intermediate resonator structure to a second resonator structure, wherein the coupling rate between the intermediate resonator structure and the second resonator structure is κ<sub>B2</sub>; and iii) during the wireless energy transfers, adjusting at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>to reduce energy accumulation in the intermediate resonator structure and improve wireless energy transfer from the first resonator structure to the second resonator structure through the intermediate resonator structure.
Embodiments of the method may include one or more of the following features.
The adjustment of at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can be selected to minimize energy accumulation in the intermediate resonator structure and cause wireless energy transfer from the first resonator structure to the second resonator structure.
The adjustment of at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can be selected to maintain energy distribution in the field of the threeresonator system in an eigenstate having substantially no energy in the intermediate resonator structure. For example, the adjustment of at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can further cause the eigenstate to evolve substantially adiabatically from an initial state with substantially all energy in the resonator structures in the first resonator structure to a final state with substantially all of the energy in the resonator structures in the second resonator structure.
The adjustment of at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can be selected to include adjustments of both coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>during wireless energy transfer.
The resonator structures can each have a quality factor larger than 10.
The first and second resonator structures can each have a quality factor greater than 50.
The first and second resonator structures can each have a quality factor greater than 100.
The resonant energy in each of the resonator structures can include electromagnetic fields. For example, the maximum value of the coupling rate κ<sub>1B </sub>and the maximum value of the coupling rate κ<sub>B2 </sub>for inductive coupling between the intermediate resonator structure and each of the first and second resonator structures can each be larger than twice the loss rate Γ for each of the first and second resonators. Moreover, The maximum value of the coupling rate κ<sub>1B </sub>and the maximum value of the coupling rate κ<sub>B2 </sub>for inductive coupling between the intermediate resonator structure and each of the first and second resonator structures can each be larger than four (4) times the loss rate Γ for each of the first and second resonators.
Each resonator structure can have a resonant frequency between 50 KHz and 500 MHz.
The maximum value of the coupling rate κ<sub>1B </sub>and the maximum value of the coupling rate κ<sub>B2 </sub>can each be at least five (5) times greater than the coupling rate between the first resonator structure and the second resonator structure.
The intermediate resonator structure can have a rate of radiative energy loss that is at least twenty (20) times greater than that for either the first resonator structure or the second resonator structure.
The first and second resonator structures can be substantially identical.
The adjustment of at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can be selected to cause peak energy accumulation in the intermediate resonator structure to be less than five percent (5%) of the peak total energy in the three resonator structures.
The adjustment of at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can be selected to cause peak energy accumulation in the intermediate resonator structure during the wireless energy transfers to be less than ten percent (10%) of the peak total energy in the three resonator structures.
Adjusting at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can include adjusting a relative position and/or orientation between one or more pairs of the resonator structures. Moreover, adjusting at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can include adjusting a resonator property of one or more of the resonator structures, such as mutual inductance.
The resonator structures can include a capacitively loaded loop or coil of at least one of a conducting wire, a conducting Litz wire, and a conducting ribbon.
The resonator structures can include an inductively loaded rod of at least one of a conducting wire, a conducting Litz wire, and a conducting ribbon.
The wireless energy transfers are nonradiative energy transfers mediated by a coupling of a resonant field evanescent tail of the first resonator structure and a resonant field evanescent tail of the intermediate resonator structure and a coupling of the resonant field evanescent tail of the intermediate resonator structure and a resonant field evanescent tail of the second resonator structure.
The adjustment of the at least one of the coupling rates can define a first mode of operation, wherein the reduction in the energy accumulation in the intermediate resonator structure is relative to energy accumulation in the intermediate resonator structure for a second mode of operation of wireless energy transfer among the three resonator structures having a coupling rate κ′<sub>1B </sub>for wireless energy transfer from the first resonator structure to the intermediate resonator structure and a coupling rate κ′<sub>B2 </sub>for wireless energy transfer from the intermediate resonator structure to the second resonator structure with κ′<sub>1B </sub>and κ′<sub>B2 </sub>each being substantially constant during the second mode of wireless energy transfer, and wherein the adjustment of the coupling rates κ<sub>1B </sub>and κ<sub>2B </sub>in the first mode of operation can be selected to κ<sub>1B</sub>, κ<sub>B2</sub><√{square root over ((κ′<sub>B2</sub><sup>2</sup>+κ′<sub>B2</sub><sup>2</sup>)/2)}. Moreover, the first mode of operation can have a greater efficiency of energy transferred from the first resonator to the second resonator compared to that for the second mode of operation. Further, the first and second resonator structures can be substantially identical and each one can have a loss rate Γ<sub>A</sub>, the intermediate resonator structure can have a loss rate Γ<sub>B</sub>, and wherein Γ<sub>B</sub>/Γ<sub>A </sub>can be greater than 50.
Also, a ratio of energy lost to radiation and total energy wirelessly transferred between the first and second resonator structures in the first mode of operation is less than that for the second mode of operation. Moreover, the first and second resonator structures can be substantially identical and each one can have a loss rate Γ<sub>A </sub>and a loss rate only due to radiation Γ<sub>A,rad</sub>, the intermediate resonator structure can have a loss rate Γ<sub>B </sub>and a loss rate only due to radiation Γ<sub>B,rad </sub>and wherein Γ<sub>B,rad</sub>/Γ<sub>B</sub>>Γ<sub>A,rad</sub>/Γ<sub>A</sub>.
The first mode of operation the intermediate resonator structure interacts less with extraneous objects than it does in the second mode of operation.
During the wireless energy transfer from the first resonator structure to the second resonator structure at least one of the coupling rates can be adjusted so that κ<sub>1B</sub><<κ<sub>B2 </sub>at a start of the energy transfer and κ<sub>1B</sub>>>κ<sub>B2 </sub>by a time a substantial portion of the energy has been transferred from the first resonator structure to the second resonator structure.
The coupling rate κ<sub>B2 </sub>can be maintained at a fixed value and the coupling rate κ<sub>1B </sub>is increased during the wireless energy transfer from the first resonator structure to second resonator structure.
The coupling rate κ<sub>1B </sub>can be maintained at a fixed value and the coupling rate κ<sub>B2 </sub>is decreased during the wireless energy transfer from the first resonator structure to second resonator structure.
During the wireless energy transfer from the first resonator structure to second resonator structure, the coupling rate κ<sub>1B </sub>can be increased and the coupling rate κ<sub>B2 </sub>is decreased.
The method may further include features corresponding to those listed for one or more of the apparatuses and methods described below.
In another aspect, disclosed is an apparatus including: first, intermediate, and second resonator structures, wherein a coupling rate between the first resonator structure and the intermediate resonator structure is κ<sub>1B </sub>and a coupling rate between the intermediate resonator structure and the second resonator structure is κ<sub>B2</sub>; and means for adjusting at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>during wireless energy transfers among the resonator structures to reduce energy accumulation in the intermediate resonator structure and improve wireless energy transfer from the first resonator structure to the second resonator structure through the intermediate resonator structure.
Embodiments for the apparatus can include one or more of the following features.
The means for adjusting can include a rotation stage for adjusting the relative orientation of the intermediate resonator structure with respect to the first and second resonator structures.
The means for adjusting can include a translation stage for moving the first and/or second resonator structures relative to the intermediate resonator structure.
The means for adjusting can include a mechanical, electromechanical, or electrical staging system for dynamically adjusting the effective size of one or more of the resonator structures.
The apparatus may further include features corresponding to those listed for the method described above, and one or more of the apparatuses and methods described below.
In another aspect, a method for transferring energy wirelessly includes i): transferring energy wirelessly from a first resonator structure to a intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ<sub>1B</sub>; ii) transferring energy wirelessly from the intermediate resonator structure to a second resonator, wherein the coupling rate between the intermediate resonator structure and the second resonator structure is κ<sub>B2</sub>; and iii) during the wireless energy transfers, adjusting at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>to cause an energy distribution in the field of the threeresonator system to have substantially no energy in the intermediate resonator structure while wirelessly transferring energy from the first resonator structure to the second resonator structure through the intermediate resonator structure.
Embodiments for the method above can include one or more of the following features.
Having substantially no energy in the intermediate resonator structure can mean that peak energy accumulation in the intermediate resonator structure is less than ten percent (10%) of the peak total energy in the three resonator structures throughout the wireless energy transfer.
Having substantially no energy in the intermediate resonator structure can mean that peak energy accumulation in the intermediate resonator structure is less than five percent (5%) of the peak total energy in the three resonator structures throughout the wireless energy transfer.
The adjustment of at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can be selected to maintain the energy distribution in the field of the threeresonator system in an eigenstate having the substantially no energy in the intermediate resonator structure.
The adjustment of at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can be selected to further cause the eigenstate to evolve substantially adiabatically from an initial state with substantially all energy in the resonator structures in the first resonator structure to a final state with substantially all of the energy in the resonator structures in the second resonator structure.
The adjustment of at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can include adjustments of both coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>during wireless energy transfers.
The resonant energy in each of the resonator structures comprises electromagnetic fields. For example, the maximum value of the coupling rate κ<sub>1B </sub>and the maximum value of the coupling rate κ<sub>B2 </sub>for inductive coupling between the intermediate resonator structure and each of the first and second resonator structures can each be larger than twice the loss rate Γ for each of the first and second resonators. Moreover, the maximum value of the coupling rate κ<sub>B </sub>and the maximum value of the coupling rate κ<sub>B2 </sub>for inductive coupling between the intermediate resonator structure and each of the first and second resonator structures can each be larger than four (4) times the loss rate Γ for each of the first and second resonators.
The resonator structure can have a resonant frequency between 50 KHz and 500 MHz.
The maximum value of the coupling rate κ<sub>1B </sub>and the maximum value of the coupling rate κ<sub>B2 </sub>can each be at least five (5) times greater than the coupling rate between the first resonator structure and the second resonator structure.
The intermediate resonator structure can have a rate of radiative energy loss that is at least twenty (20) times greater than that for either the first resonator structure or the second resonator structure.
The first and second resonator structures can be substantially identical.
Adjusting at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can include adjusting a relative position and/or orientation between one or more pairs of the resonator structures.
Adjusting at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can include adjusting a resonator property of one or more of the resonator structures, such as mutual inductance.
The resonator structures can include a capacitively loaded loop or coil of at least one of a conducting wire, a conducting Litz wire, and a conducting ribbon.
The resonator structures can include an inductively loaded rod of at least one of a conducting wire, a conducting Litz wire, and a conducting ribbon.
The wireless energy transfers can be nonradiative energy transfers mediated by a coupling of a resonant field evanescent tail of the first resonator structure and a resonant field evanescent tail of the intermediate resonator structure and a coupling of the resonant field evanescent tail of the intermediate resonator structure and a resonant field evanescent tail of the second resonator structure.
The first and second resonator structures can each have a quality factor greater than 50.
The first and second resonator structures can each have a quality factor greater than 100.
The adjustment of at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>can be selected to cause the energy distribution in the field of the threeresonator system to have substantially no energy in the intermediate resonator structure improves wireless energy transfer between the first and second resonator structures.
The adjustment of the at least one of the coupling rates can be selected to define a first mode of operation, wherein energy accumulation in the intermediate resonator structure during the wireless energy transfer from the first resonator structure to second resonator structure is smaller than that for a second mode of operation of wireless energy transfer among the three resonator structures having a coupling rate κ′<sub>1B </sub>for wireless energy transfer from the first resonator structure to the intermediate resonator structure and a coupling rate κ′<sub>B2 </sub>for wireless energy transfer from the intermediate resonator structure to the second resonator structure with κ′<sub>1B </sub>and κ′<sub>B2 </sub>each being substantially constant during the second mode of wireless energy transfer, and wherein the adjustment of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>in the first mode of operation can be selected to satisfy κ<sub>1B</sub>, κ<sub>B2</sub><√{square root over ((κ′<sub>B2</sub><sup>2</sup>+κ′<sub>B2</sub><sup>2</sup>)/2)}.
The first mode of operation can have a greater efficiency of energy transferred from the first resonator to the second resonator compared to that for the second mode of operation.
The first and second resonator structures can be substantially identical and each one can have a loss rate Γ<sub>A</sub>, the intermediate resonator structure can have a loss rate Γ<sub>B</sub>, and wherein Γ<sub>B</sub>/Γ<sub>A </sub>can be greater than 50.
A ratio of energy lost to radiation and total energy wirelessly transferred between the first and second resonator structures in the first mode of operation can be less than that for the second mode of operation.
The first and second resonator structures can be substantially identical and each one can have a loss rate Γ<sub>A </sub>and a loss rate only due to radiation Γ<sub>A,rad</sub>, the intermediate resonator structure can have a loss rate Γ<sub>B </sub>and a loss rate only due to radiation Γ<sub>B,rad </sub>and wherein Γ<sub>B,rad</sub>/Γ<sub>B</sub>>Γ<sub>A,rad</sub>/Γ<sub>A</sub>.
The first mode of operation the intermediate resonator structure interacts less with extraneous objects than it does in the second mode of operation.
During the wireless energy transfer from the first resonator structure to the second resonator structure at least one of the coupling rates can be adjusted so that κ<sub>1B</sub><<κ<sub>B2 </sub>at a start of the energy transfer and κ<sub>1B</sub>>>κ<sub>B2 </sub>by a time a substantial portion of the energy has been transferred from the first resonator structure to the second resonator structure.
The coupling rate κ<sub>B2 </sub>can be maintained at a fixed value and the coupling rate κ<sub>1B </sub>can be increased during the wireless energy transfer from the first resonator structure to second resonator structure.
The coupling rate κ<sub>1B </sub>can be maintained at a fixed value and the coupling rate κ<sub>B2 </sub>can be decreased during the wireless energy transfer from the first resonator structure to second resonator structure.
During the wireless energy transfer from the first resonator structure to second resonator structure, the coupling rate κ<sub>1B </sub>can be increased and the coupling rate κ<sub>B2 </sub>can be decreased.
The method may further include features corresponding to those listed for the apparatus and method described above, and one or more of the apparatuses and methods described below.
In another aspect, disclosed is an apparatus including: first, intermediate, and second resonator structures, wherein a coupling rate between the first resonator structure and the intermediate resonator structure is κ<sub>1B </sub>and a coupling rate between the intermediate resonator structure and the second resonator structure is κ<sub>B2</sub>; and means for adjusting at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>during wireless energy transfers among the resonator structures to cause an energy distribution in the field of the threeresonator system to have substantially no energy in the intermediate resonator structure while wirelessly transferring energy from the first resonator structure to the second resonator structure through the intermediate resonator structure.
Embodiments for the apparatus can include one or more of the following features.
Having substantially no energy in the intermediate resonator structure can mean that peak energy accumulation in the intermediate resonator structure is less than ten percent (10%) of the peak total energy in the three resonator structures throughout the wireless energy transfers.
Having substantially no energy in the intermediate resonator structure can mean that peak energy accumulation in the intermediate resonator structure is less than five percent (5%) of the peak total energy in the three resonator structures throughout the wireless energy transfers.
The means for adjusting can be configured to maintain the energy distribution in the field of the threeresonator system in an eigenstate having the substantially no energy in the intermediate resonator structure.
The means for adjusting can include a rotation stage for adjusting the relative orientation of the intermediate resonator structure with respect to the first and second resonator structures.
The means for adjusting can include a translation stage for moving the first and/or second resonator structures relative to the intermediate resonator structure.
The means for adjusting can include a mechanical, electromechanical, or electrical staging system for dynamically adjusting the effective size of one or more of the resonator structures.
The resonator structures can include a capacitively loaded loop or coil of at least one of a conducting wire, a conducting Litz wire, and a conducting ribbon.
The resonator structures can include an inductively loaded rod of at least one of a conducting wire, a conducting Litz wire, and a conducting ribbon.
A source can be coupled to the first resonator structure and a load can be coupled to the second resonator structure.
The apparatus may further include features corresponding to those listed for the apparatus and methods described above, and the apparatus and method described below.
In another aspect, disclosed is a method for transferring energy wirelessly that includes: i) transferring energy wirelessly from a first resonator structure to a intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ<sub>1B</sub>; ii) transferring energy wirelessly from the intermediate resonator structure to a second resonator, wherein the coupling rate between the intermediate resonator structure and the second resonator structure with a coupling rate is κ<sub>B2</sub>; and iii) during the wireless energy transfers, adjusting at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>to define a first mode of operation in which energy accumulation in the intermediate resonator structure is reduced relative to that for a second mode of operation of wireless energy transfer among the three resonator structures having a coupling rate κ<sub>1B </sub>for wireless energy transfer from the first resonator structure to the intermediate resonator structure and a coupling rate κ′<sub>B2 </sub>for wireless energy transfer from the intermediate resonator structure to the second resonator structure with κ′<sub>1B </sub>and κ′<sub>B2 </sub>each being substantially constant during the second mode of wireless energy transfer, and wherein the adjustment of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>in the first mode of operation can be selected to satisfy κ<sub>1B</sub>, κ<sub>B2</sub><√{square root over ((κ′<sub>B2</sub><sup>2</sup>+κ′<sub>B2</sub><sup>2</sup>)/2)}.
The method may further include features corresponding to those listed for the apparatuses and methods described above.
In another aspect, disclosed is an apparatus that includes: first, intermediate, and second resonator structures, wherein a coupling rate between the first resonator structure and the intermediate resonator structure is κ<sub>B </sub>and a coupling rate between the intermediate resonator structure and the second resonator structure is κ<sub>B2</sub>; and means for adjusting at least one of the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>during wireless energy transfers among the resonator structures to define a first mode of operation in which energy accumulation in the intermediate resonator structure is reduced relative to that for a second mode of operation for wireless energy transfer among the three resonator structures having a coupling rate κ′<sub>1B </sub>for wireless energy transfer from the first resonator structure to the intermediate resonator structure and a coupling rate κ′<sub>B2 </sub>for wireless energy transfer from the intermediate resonator structure to the second resonator structure with κ<sub>1B </sub>and κ′<sub>B2 </sub>each being substantially constant during the second mode of wireless energy transfer, and wherein the adjustment of the coupling rates κ<sub>12 </sub>and κ<sub>B2 </sub>in the first mode of operation can be selected to satisfy κ<sub>1B</sub>, κ<sub>B2</sub><√{square root over ((κ′<sub>B2</sub><sup>2</sup>+κ′<sub>B2</sub><sup>2</sup>)/2)}.
The apparatus may further include features corresponding to those listed for the apparatuses and methods described above.
Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art. Although methods and materials similar or equivalent to those described herein can be used in the practice or testing of the present disclosure, suitable methods and materials are described below. All publications, patent applications, patents, and other references mentioned herein are incorporated by reference in their entirety. In case of conflict, the present specification, including definitions, will control. In addition, the materials, methods, and examples are illustrative only and not intended to be limiting.
The details of one or more embodiments are set forth in the accompanying drawings and the description below, including the documents appended hereto. Other features and advantages will be apparent from this disclosure and from the claims.
BRIEF DESCRIPTION OF THE DRAWINGSFIG. 1 shows a schematic of an example wireless energy transfer scheme.
FIGS. 2(a)(b) show the efficiency of power transmission η<sub>P </sub>for (a) U=1 and (b) U=3, as a function of the frequency detuning D<sub>o </sub>and for different values of the loading rate U<sub>o</sub>.
FIG. 2(c) shows the optimal (for zero detuning and under conditions of impedance matching) efficiency for energy transfer η<sub>E* </sub>and power transmission η<sub>P*</sub>, as a function of the couplingtoloss figureofmerit U.
FIG. 3 shows an example of a selfresonant conductingwire coil.
FIG. 4 shows an example of a wireless energy transfer scheme featuring two selfresonant conductingwire coils.
FIG. 5 is a schematic of an experimental system demonstrating wireless energy transfer.
FIG. 6 shows a comparison between experimental and theoretical results for the coupling rate of the system shown schematically in FIG. 5.
FIG. 7 shows a comparison between experimental and theoretical results for the strongcoupling factor of the system shown schematically in FIG. 5.
FIG. 8 shows a comparison between experimental and theoretical results for the powertransmission efficiency of the system shown schematically in FIG. 5.
FIG. 9 shows an example of a capacitivelyloaded conductingwire coil, and illustrates the surrounding field.
FIG. 10 shows an example wireless energy transfer scheme featuring two capacitivelyloaded conductingwire coils, and illustrates the surrounding field.
FIG. 11 illustrates an example circuit model for wireless energy transfer.
FIG. 12 shows the efficiency, total (loaded) device Q, and source and device currents, voltages and radiated powers (normalized to 1 Watt of output power to the load) as functions of the resonant frequency, for a particular choice of source and device loop dimensions, wp and N<sub>s </sub>and different choices of N<sub>d</sub>=1, 2, 3, 4, 5, 6, 10.
FIG. 13 shows the efficiency, total (loaded) device Q, and source and device currents, voltages and radiated powers (normalized to 1 Watt of output power to the load) as functions of frequency and wp for a particular choice of source and device loop dimensions, and number of turns N<sub>s </sub>and N<sub>d</sub>.
FIG. 14 shows an example of an inductivelyloaded conductingwire coil.
FIG. 15 shows (a) an example of a resonant dielectric disk, and illustrates the surrounding field and (b) a wireless energy transfer scheme featuring two resonant dielectric disks, and illustrates the surrounding field.
FIG. 16 shows a schematic of an example wireless energy transfer scheme with one source resonator and one device resonator exchanging energy indirectly through an intermediate resonator.
FIG. 17 shows an example of a wireless energy transfer system: (a) (Left) Schematic of loops configuration in twoobject direct transfer. (Right) Time evolution of energies in the twoobject direct energy transfer case. (b) (Left) Schematic of threeloops configuration in the constantκ case. (Right) Dynamics of energy transfer for the configuration in (b. Left). Note that the total energy transferred E<sub>2 </sub>is 2 times larger than in (a. Right), but at the price of the total energy radiated being 4 times larger. (c) (Left) Loop configuration at t=0 in the adiabaticκ scheme. (Center) Dynamics of energy transfer with adiabatically rotating loops. (Right) Loop configuration at t=t<sub>EIT</sub>. Note that E<sub>2 </sub>is comparable to (b. Right), but the radiated energy is now much smaller: In fact, it is comparable to (a. Right).
FIG. 18 shows a schematic of an example wireless energy transfer scheme with one source resonator and one device resonator exchanging energy indirectly through an intermediate resonator, where an adjustment system is used to rotate the resonator structures to dynamically adjust their coupling rates.
FIG. 19 shows an example of a temporal variation of the coupling rates in a wireless energy transfer system as in FIG. 18 to achieve an adiabatic transfer of energy from the source object R<sub>1 </sub>to the device object R<sub>2</sub>.
FIG. 20 shows the energy distribution in a wireless energy transfer system as in FIG. 18 as a function of time when the coupling rates are timevarying, for Γ<sub>A</sub>=0, κ/Γ<sub>B</sub>=10, κ<sub>1B</sub>=κ sin [πt/(2t<sub>EΓΓ</sub>)], and κ<sub>B2</sub>=κ cos [πt/(2t<sub>EΓΓ</sub>)].
FIGS. 21(a)(f) show a comparison between the adiabaticκ and constantκ energy transfer schemes, in the general case: (a) Optimum E<sub>2 </sub>(%) in adiabaticκ transfer, (b) Optimum E<sub>2 </sub>(%) in constantκ transfer, (c) (E<sub>2</sub>)<sub>adiabaticκ</sub>/(E<sub>2</sub>)<sub>constantκ</sub>, (d) Energy lost (%) at optimum adiabaticκ transfer, (e) Energy lost (%) at optimum constantκ transfer, (f) (E<sub>lost</sub>)<sub>constantκ</sub>/(E<sub>lost</sub>)<sub>adiabaticκ</sub>.
FIG. 22(a)(e) show a comparison between radiated energies in the adiabaticκ and constantκ energy transfer schemes: (a) E<sub>rad</sub>(%) in the constantscheme for Γ<sub>B</sub>/Γ<sub>A</sub>=500 and Γ<sub>rad</sub><sup>A</sup>=0, (b) E<sub>rad</sub>(%) in the adiabaticκ scheme for Γ<sub>B</sub>/Γ<sub>A</sub>=500 and Γ<sub>rad</sub><sup>A</sup>=0, (c) (E<sub>rad</sub>) constantκ/(E<sub>rad</sub>)<sub>adiabaticκ</sub> for Γ<sub>B</sub>/Γ<sub>A</sub>=50, (d) (E<sub>rad</sub>)<sub>constantκ</sub>/(E<sub>rad</sub>)<sub>adiabaticκ</sub> for Γ<sub>B</sub>/r<sub>A</sub>=500, (e) [(E<sub>rad</sub>)<sub>constantκ</sub>/(E<sub>rad</sub>)<sub>adiabaticκ</sub>] as function of κ/Γ<sub>B </sub>and Γ<sub>B</sub>/Γ<sub>A</sub>, forΓ<sub>rad</sub><sup>A</sup>=0.
FIGS. 23(a)(b) show schematics for frequency control mechanisms.
FIGS. 24(a)(c) illustrate a wireless energy transfer scheme using two dielectric disks in the presence of various extraneous objects.
DETAILED DESCRIPTIONEfficient wireless energytransfer between two similarfrequency resonant objects can be achieved at midrange distances, provided these resonant objects are designed to operate in the ‘strongcoupling’ regime. ‘Strong coupling’ can be realized for a wide variety of resonant objects, including electromagnetic resonant objects such as inductivelyloaded conducting rods and dielectric disks. Recently, we have demonstrated wireless energy transfer between strongly coupled electromagnetic selfresonant conducting coils and capacitivelyloaded conducting coils, bearing highQ electromagnetic resonant modes. See, for example, the following commonly owned U.S. patent applications, all of which are incorporated herein by reference: U.S. application Ser. No. 11/481,077, filed on Jul. 5, 2006, and published as U.S. Patent Publication No. US 20070222542 A1; U.S. application Ser. No. 12/055,963, filed on Mar. 26, 2008, and published as U.S. Patent Publication No. US 20080278264 A1; and U.S. patent application Ser. No. 12/466,065, filed on May 14, 2009, and published as U.S. patent Ser. No. ______. In general, the energytransfer efficiency between similarfrequency, strongly coupled resonant objects decreases as the distance between the objects is increased.
In this work, we explore a further scheme of efficient energy transfer between resonant objects that extends the range over which energy may be efficiently transferred. Instead of transferring energy directly between two resonant objects, as has been described in certain embodiments of the crossreferenced patents, in certain embodiments, an intermediate resonant object, with a resonant frequency equal or nearlyequal to that of the two energyexchanging resonant objects is used to mediate the transfer. The intermediate resonant object may be chosen so that it couples more strongly to each of the resonant objects involved in the energy transfer than those two resonant objects couple to each other. One way to design such an intermediate resonator is to make it larger than either of the resonant objects involved in the energy transfer. However, increasing the size of the intermediate resonant object may lower its quality factor, or Q, by increasing its radiation losses. Surprisingly enough, this new “indirect” energy transfer scheme may be shown to be very efficient and only weaklyradiative by introducing a meticulously chosen time variation of the resonator coupling rates.
The advantage of this method over the prior commonly owned wireless energy transfer techniques is that, in certain embodiments, it can enable energy to be transferred wirelessly between two objects with a larger efficiency and/or with a smaller radiation loss and/or with fewer interactions with extraneous objects.
Accordingly, in certain embodiments, we disclose an efficient wireless energy transfer scheme between two similar resonant objects, strongly coupled to an intermediate resonant object of substantially different properties, but with the same resonance frequency. The transfer mechanism essentially makes use of the adiabatic evolution of an instantaneous (so called ‘dark’) resonant state of the coupled threeobject system. Our analysis is based on temporal coupled mode theory (CMT), and is general. Of particular commercial interest is the application of this technique to stronglycoupled electromagnetic resonators used for midrange wireless energy transfer applications. We show that in certain parameter regimes of interest, this scheme can be more efficient, and/or less radiative than other wireless energy transfer approaches.
While the technique described herein is primarily directed to tangible resonator structures, the technique shares certain features with a quantum interference phenomenon known in the atomic physics community as Electromagnetically Induced Transparency (EIT). In EIT, three atomic states participate. Two of them, which are nonlossy, are coupled to one that has substantial losses. However, by meticulously controlling the mutual couplings between the states, one can establish a coupled system which is overall nonlossy. This phenomena has been demonstrated using carefully timed optical pulses, referred to as probe laser pulses and Stokes laser pulses, to reduce the opacity of media with the appropriate collection of atomic states. A closely related phenomenon known as Stimulated Raman Adiabatic Passage (STIRAP) may take place in a similar system; namely, the probe and Stokes laser beams may be used to achieve complete coherent population transfer between two molecular states of a medium. Hence, we may refer to the currently proposed scheme as the “EITlike” energy transfer scheme.
In certain embodiments, we disclose an efficient nearfield energy transfer scheme between two similar resonant objects, based on an EITlike transfer of the energy through a mediating resonant object with the same resonant frequency. In embodiments, this EITlike energy transfer may be realized using electromagnetic resonators as have been described in the crossreferenced patents, but the scheme is not bound only to wireless energy transfer applications. Rather, this scheme is general and may find applications in various other types of coupling between general resonant objects. In certain embodiments described below, we describe particular examples of electromagnetic resonators, but the nature of the resonators and their coupling mechanisms could be quite different (e.g. acoustic, mechanical, etc.). To the extent that many resonant phenomena can be modeled with nearly identical CMT equations, similar behavior to that described herein would occur.
FIG. 1 shows a schematic that generally describes one example of the invention, in which energy is transferred wirelessly between two resonant objects. Referring to FIG. 1, energy is transferred over a distance D, between a resonant source object having a characteristic size r<sub>1 </sub>and a resonant device object of characteristic size r<sub>2</sub>. Both objects are resonant objects. The wireless nearfield energy transfer is performed using the field (e.g. the electromagnetic field or acoustic field) of the system of two resonant objects.
The characteristic size of an object can be regarded as being equal to the radius of the smallest sphere which can fit around the entire object. The characteristic thickness of an object can be regarded as being, when placed on a flat surface in any arbitrary configuration, the smallest possible height of the highest point of the object above a flat surface. The characteristic width of an object can be regarded as being the radius of the smallest possible circle that the object can pass through while traveling in a straight line. For example, the characteristic width of a cylindrical object is the radius of the cylinder.
Initially, we present a theoretical framework for understanding nearfield wireless energy transfer. Note however that it is to be understood that the scope of the invention is not bound by theory.
Different temporal schemes can be employed, depending on the application, to transfer energy between two resonant objects. Here we will consider two particularly simple but important schemes: a onetime finiteamount energytransfer scheme and a continuous finiterate energytransfer (power) scheme.
Let the source and device objects be 1, 2 respectively and their resonance modes, which we will use for the energy exchange, have angular frequencies ω<sub>1,2 </sub>frequencywidths due to intrinsic (absorption, radiation etc.) losses Γ<sub>1,2 </sub>and (generally) vector fields F<sub>1,2</sub>(r), normalized to unity energy. Once the two resonant objects are brought in proximity, they can interact and an appropriate analytical framework for modeling this resonant interaction is that of the wellknown coupledmode theory (CMT). This model works well, when the resonances are well defined by having large quality factors and their resonant frequencies are relatively close to each other. In this picture, the field of the system of the two resonant objects 1, 2 can be approximated by F(r,t)=(t)F<sub>1</sub>(r)+a<sub>2</sub>(t)F<sub>2</sub>(r), where a<sub>1,2 </sub>(t) are the field amplitudes, with a<sub>1,2</sub>(t)<sup>2 </sup>equal to the energy stored inside the object 1, 2 respectively, due to the normalization. Then, using e<sup>−iωt </sup>time dependence, the field amplitudes can be shown to satisfy, to lowest order:
<maths id="MATHUS00001" num="00001"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mfrac><mo></mo><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo></mo><mrow><mi></mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>ω</mi><mn>1</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>11</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>12</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo></mo><mrow><mrow><mfrac><mo></mo><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo></mo><mrow><mi></mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>ω</mi><mn>2</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mn>2</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>21</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>22</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where κ<sub>11,22 </sub>are the shifts in each object's frequency due to the presence of the other, which are a secondorder correction and can be absorbed into the resonant frequencies (eigenfrequencies) by setting ω<sub>1,2</sub>→ω<sub>1,2</sub>+κ<sub>11,22</sub>, and κ<sub>12,21 </sub>are the coupling coefficients, which from the reciprocity requirement of the system satisfy κ<sub>21</sub>=κ<sub>12</sub>≡κ.
The resonant modes of the combined system are found by substituting [a<sub>1</sub>(t), a<sub>2</sub>(t)]=[A<sub>1</sub>, A<sub>2</sub>]e<sup>−i <o ostyle="single">ω</o>t</sup>. They have complex resonant frequencies
<FORM><o ostyle="single">ω</o><sub>±</sub>=ω<sub>12</sub>±√{square root over ((Δω<sub>12</sub>)<sup>2</sup>+κ<sup>2</sup>)} (2a)</FORM>
where ω<sub>12</sub>=[(ω<sub>1</sub>+ω<sub>2</sub>)−i(Γ<sub>1</sub>+Γ<sub>2</sub>)]/2, Δω<sub>12</sub>−[(ω<sub>1</sub>−ω<sub>2</sub>)−i(Γ<sub>1</sub>−Γ<sub>2</sub>)]/2 and whose splitting we denote as δ<sub>E</sub>≡ <o ostyle="single">ω</o><sub>+</sub>− <o ostyle="single">ω</o><sub>−</sub>, and corresponding resonant field amplitudes
<maths id="MATHUS00002" num="00002"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mover><mi>V</mi><mo>></mo></mover><mo>±</mo></msub><mo>=</mo><mrow><msub><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>A</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>±</mo></msub><mo>=</mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd><mi>κ</mi></mtd></mtr><mtr><mtd><mrow><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>ω</mi><mn>12</mn></msub></mrow><mo>∓</mo><msqrt><mrow><msup><mrow><mo>(</mo><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>ω</mi><mn>12</mn></msub></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>κ</mi><mn>2</mn></msup></mrow></msqrt></mrow></mtd></mtr></mtable><mo>]</mo></mrow><mo>.</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
Note that, at exact resonance ω<sub>1</sub>=ω<sub>2</sub>=ω<sub>A </sub>and for Γ<sub>1</sub>=Γ<sub>2</sub>=Γ<sub>A</sub>, we get Δω<sub>12</sub>=0, δ<sub>E</sub>=2K, and then
<maths id="MATHUS00003" num="00003"><math overflow="scroll"><mrow><msub><mover><mi>ω</mi><mi>_</mi></mover><mo>±</mo></msub><mo>=</mo><mrow><mrow><msub><mi>ω</mi><mi>A</mi></msub><mo>±</mo><mi>κ</mi></mrow><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mi>A</mi></msub></mrow></mrow></mrow></math></maths><maths id="MATHUS000032" num="00003.2"><math overflow="scroll"><mrow><mrow><msub><mover><mi>V</mi><mo>></mo></mover><mo>±</mo></msub><mo>=</mo><mrow><msub><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>A</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>±</mo></msub><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mrow><mo>∓</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mrow><mo>,</mo></mrow></math></maths>
namely we get the known result that the resonant modes split to a lower frequency even mode and a higher frequency odd mode.
Assume now that at time t=0 the source object 1 has finite energy a<sub>1</sub>(0)<sup>2</sup>, while the device object has a<sub>2</sub>(0)<sup>2</sup>=0. Since the objects are coupled, energy will be transferred from 1 to 2. With these initial conditions, Eqs. (1) can be solved, predicting the evolution of the device fieldamplitude to be
<maths id="MATHUS00004" num="00004"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mrow><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo></mo></mrow></mfrac><mo>=</mo><mrow><mfrac><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>κ</mi></mrow><msub><mi>δ</mi><mi>E</mi></msub></mfrac><mo></mo><mrow><mi>sin</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><msub><mi>δ</mi><mi>E</mi></msub><mo></mo><mi>t</mi></mrow><mn>2</mn></mfrac><mo>)</mo></mrow></mrow><mo></mo><msup><mi></mi><mrow><mrow><mo></mo><mfrac><mrow><msub><mi>Γ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>Γ</mi><mn>2</mn></msub></mrow><mn>2</mn></mfrac></mrow><mo></mo><mi>t</mi></mrow></msup><mo></mo><mrow><msup><mi></mi><mrow><mrow><mo></mo><mi></mi></mrow><mo></mo><mfrac><mrow><msub><mi>ω</mi><mn>1</mn></msub><mo>+</mo><msub><mi>ω</mi><mn>2</mn></msub></mrow><mn>2</mn></mfrac><mo></mo><mi>t</mi></mrow></msup><mo>.</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
The energytransfer efficiency will be η<sub>E</sub>≡a<sub>2</sub>(t)<sup>2</sup>/a<sub>1</sub>(0)<sup>2</sup>. The ratio of energy converted to loss due to a specific loss mechanism in resonators 1 and 2, with respective loss rates Γ<sub>1,loss </sub>and Γ<sub>2,loss </sub>will be η<sub>loss,E</sub>=∫<sub>0</sub><sup>t</sup>dτ[2Γ<sub>1,loss</sub>a<sub>1</sub>(τ)<sup>2</sup>+2Γ<sub>2,loss</sub>a<sub>2</sub>(τ)<sup>2</sup>]/a<sub>1</sub>(0)<sup>2</sup>.Note that, at exact resonance ω<sub>1</sub>=ω<sub>2</sub>=ω<sub>A </sub>(an optimal condition), Eq. (3) can be written as
<maths id="MATHUS00005" num="00005"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow><mrow><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow><mo></mo></mrow></mfrac><mo>=</mo><mrow><mfrac><mrow><mi>sin</mi><mo></mo><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mi>Δ</mi><mn>2</mn></msup></mrow></msqrt><mo></mo><mi>T</mi></mrow><mo>)</mo></mrow></mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mi>Δ</mi><mn>2</mn></msup></mrow></msqrt></mfrac><mo></mo><msup><mi></mi><mrow><mrow><mo></mo><mi>T</mi></mrow><mo>/</mo><mi>U</mi></mrow></msup><mo></mo><msup><mi></mi><mrow><mrow><mo></mo><mi></mi></mrow><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>ω</mi><mi>A</mi></msub><mo></mo><mi>t</mi></mrow></msup></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where ≡κt, Δ<sup>−1</sup>=2κ/(Γ<sub>2</sub>−Γ<sub>1</sub>) and U=2κ/(Γ<sub>1</sub>+Γ<sub>2</sub>).
In some examples, the system designer can adjust the duration of the coupling t at will. In some examples, the duration t can be adjusted to maximize the device energy (and thus efficiency η<sub>E</sub>). Then, it can be inferred from Eq. (4) that η<sub>E </sub>is maximized for
<maths id="MATHUS00006" num="00006"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>T</mi><mo>*</mo></msub><mo>=</mo><mfrac><mrow><msup><mi>tan</mi><mrow><mo></mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mi>U</mi><mo></mo><msqrt><mrow><mn>1</mn><mo></mo><msup><mi>Δ</mi><mn>2</mn></msup></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mi>Δ</mi><mn>2</mn></msup></mrow></msqrt></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
resulting in an optimal energytransfer efficiency
<maths id="MATHUS00007" num="00007"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>η</mi><msup><mi>E</mi><mo>*</mo></msup></msub><mo>≡</mo><mrow><msub><mi>η</mi><mi>E</mi></msub><mo></mo><mrow><mo>(</mo><msub><mi>T</mi><mo>*</mo></msub><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><msup><mi>U</mi><mn>2</mn></msup><mrow><mn>1</mn><mo>+</mo><mrow><msup><mi>U</mi><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo></mo><msup><mi>Δ</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow></mfrac><mo></mo><mrow><mrow><mi>exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo></mo><mfrac><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mrow><msup><mi>tan</mi><mrow><mo></mo><mn>1</mn></mrow></msup><mo></mo><mrow><mo>(</mo><mrow><mi>U</mi><mo></mo><msqrt><mrow><mn>1</mn><mo></mo><msup><mi>Δ</mi><mn>2</mn></msup></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>U</mi><mo></mo><msqrt><mrow><mn>1</mn><mo></mo><msup><mi>Δ</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>.</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mrow><mn>6</mn><mo></mo><mi>a</mi></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
which is a monotonically increasing function of the couplingtoloss ratio U=2κ/(Γ<sub>1</sub>+Γ<sub>2</sub>) and tends to unity when U>>1<img id="CUSTOMCHARACTER00001" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>Δ<sup>−1</sup>>>1. Therefore, the energy transfer is nearly perfect, when the coupling rate is much faster than all loss rates (κ/Γ<sub>1,2</sub>>>1). In FIG. 2(c) we show the optimal energytransfer efficiency when Γ<sub>1</sub>=Γ<sub>2</sub>−Γ<sub>A</sub><img id="CUSTOMCHARACTER00002" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00002.TIF" imgcontent="character" imgformat="tif"/>Δ=0:
<maths id="MATHUS00008" num="00008"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>η</mi><mi>E</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi>T</mi><mo>*</mo></msub><mo>,</mo><mrow><mi>Δ</mi><mo>=</mo><mn>0</mn></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><msup><mi>U</mi><mn>2</mn></msup><mrow><mn>1</mn><mo>+</mo><msup><mi>U</mi><mn>2</mn></msup></mrow></mfrac><mo></mo><mrow><mrow><mi>exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo></mo><mfrac><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>tan</mi><mrow><mo></mo><mn>1</mn></mrow></msup><mo></mo><mi>U</mi></mrow><mi>U</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mo>.</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mrow><mn>6</mn><mo></mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
In a real wireless energytransfer system, the source object can be connected to a power generator (not shown in FIG. 1), and the device object can be connected to a power consuming load (e.g. a resistor, a battery, an actual device, not shown in FIG. 1). The generator will supply the energy to the source object, the energy will be transferred wirelessly and nonradiatively from the source object to the device object, and the load will consume the energy from the device object. To incorporate such supply and consumption mechanisms into this temporal scheme, in some examples, one can imagine that the generator is very briefly but very strongly coupled to the source at time t=0 to almost instantaneously provide the energy, and the load is similarly very briefly but very strongly coupled to the device at the optimal time t=t<sub>*</sub>, to almost instantaneously drain the energy. For a constant powering mechanism, at time t=t<sub>*</sub>, also the generator can again be coupled to the source to feed a new amount of energy, and this process can be repeated periodically with a period t<sub>*</sub>.
Let the generator be continuously supplying energy to the source object 1 at a rate κ<sub>1 </sub>and the load continuously draining energy from the device object 2 at a rate κ<sub>2</sub>. Field amplitudes s<sub>±1,2</sub>(t) are then defined, so that s<sub>±1,2</sub>(t)<sup>2 </sup>is equal to the power ingoing to (for the + sign) or outgoing from (for the − sign) the object 1, 2 respectively, and the CMT equations are modified to
<maths id="MATHUS00009" num="00009"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mfrac><mo></mo><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo></mo><mrow><mi></mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>ω</mi><mn>1</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>11</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>12</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><msub><mi>κ</mi><mn>1</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><msqrt><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>1</mn></msub></mrow></msqrt><mo></mo><mrow><msub><mi>s</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo></mo><mstyle><mspace width="4.4em" height="4.4ex"/></mstyle><mo></mo><mrow><mrow><mfrac><mo></mo><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo></mo><mrow><mi></mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>ω</mi><mn>2</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mn>2</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>21</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>22</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><msub><mi>κ</mi><mn>2</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo></mo><mstyle><mspace width="4.4em" height="4.4ex"/></mstyle><mo></mo><mrow><mrow><msub><mi>s</mi><mrow><mo></mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msqrt><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>1</mn></msub></mrow></msqrt><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><msub><mi>s</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo></mo><mstyle><mspace width="4.4em" height="4.4ex"/></mstyle><mo></mo><mrow><mrow><msub><mi>s</mi><mrow><mo></mo><mn>2</mn></mrow></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msqrt><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>2</mn></msub></mrow></msqrt><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where again we can set ω<sub>1,2</sub>→ω<sub>1,2</sub>+κ<sub>11,22 </sub>and κ<sub>21</sub>=κ<sub>12</sub>≡κ.
Assume now that the excitation is at a fixed frequency ω, namely has the form s<sub>+1</sub>(t)=S<sub>+1</sub>e<sup>−iωt</sup>. Then the response of the linear system will be at the same frequency, namely a<sub>1,2</sub>(t)=A<sub>1,2</sub>e<sup>−iωt </sup>and s<sub>−1,2</sub>(t)=S<sub>−1,2</sub><sup>−iωt</sup>. By substituting these into Eqs. (7), using δ<sub>1,2</sub>≡ω−ω<sub>1,2</sub>, and solving the system, we find the fieldamplitude transmitted to the load (S<sub>21 </sub>scatteringmatrix element)
<maths id="MATHUS00010" num="00010"><math overflow="scroll"><mtable><mtr><mtd><mtable><mtr><mtd><mrow><mrow><msub><mi>S</mi><mn>21</mn></msub><mo>≡</mo><mfrac><msub><mi>S</mi><mrow><mo></mo><mn>2</mn></mrow></msub><msub><mi>S</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub></mfrac></mrow><mo>=</mo><mi/><mo></mo><mfrac><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>κ</mi><mo></mo><msqrt><mrow><msub><mi>κ</mi><mn>1</mn></msub><mo></mo><msub><mi>κ</mi><mn>2</mn></msub></mrow></msqrt></mrow><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>Γ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>κ</mi><mn>1</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>δ</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><msub><mi>Γ</mi><mn>2</mn></msub><mo>+</mo><msub><mi>κ</mi><mn>2</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>δ</mi><mn>2</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><msup><mi>κ</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mi/><mo></mo><mfrac><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>U</mi><mo></mo><msqrt><mrow><msub><mi>U</mi><mn>1</mn></msub><mo></mo><msub><mi>U</mi><mn>2</mn></msub></mrow></msqrt></mrow><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>U</mi><mn>1</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>D</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>U</mi><mn>2</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>D</mi><mn>2</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><msup><mi>U</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
and the fieldamplitude reflected to the generator (S<sub>11 </sub>scatteringmatrix element)
<maths id="MATHUS00011" num="00011"><math overflow="scroll"><mtable><mtr><mtd><mtable><mtr><mtd><mrow><mrow><msub><mi>S</mi><mn>11</mn></msub><mo>≡</mo><mfrac><msub><mi>S</mi><mrow><mo></mo><mn>1</mn></mrow></msub><msub><mi>S</mi><mrow><mo>+</mo><mn>1</mn></mrow></msub></mfrac></mrow><mo>=</mo><mi/><mo></mo><mfrac><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>Γ</mi><mn>1</mn></msub><mo></mo><msub><mi>κ</mi><mn>1</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>δ</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><msub><mi>Γ</mi><mn>2</mn></msub><mo>+</mo><msub><mi>κ</mi><mn>2</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>δ</mi><mn>2</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><msup><mi>κ</mi><mn>2</mn></msup></mrow><mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>Γ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>κ</mi><mn>1</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>δ</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><msub><mi>Γ</mi><mn>2</mn></msub><mo>+</mo><msub><mi>κ</mi><mn>2</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>δ</mi><mn>2</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><msup><mi>κ</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mi/><mo></mo><mfrac><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo></mo><msub><mi>U</mi><mn>1</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>D</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>U</mi><mn>2</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>D</mi><mn>2</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><msup><mi>U</mi><mn>2</mn></msup></mrow><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>U</mi><mn>1</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>D</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>U</mi><mn>2</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>D</mi><mn>2</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><msup><mi>U</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>9</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where D<sub>1,2</sub>≡δ<sub>1,2</sub>/Γ<sub>1,2</sub>, U<sub>1,2</sub>≡κ<sub>1,2</sub>/Γ<sub>1,2 </sub>and U≡κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)}. Similarly, the scatteringmatrix elements S<sub>12</sub>, S<sub>22 </sub>are given by interchanging 1<img id="CUSTOMCHARACTER00003" he="1.78mm" wi="3.89mm" file="US20100148589A120100617P00003.TIF" imgcontent="character" imgformat="tif"/>2 in Eqs. (8), (9) and, as expected from reciprocity, S<sub>21</sub>=S<sub>12</sub>. The coefficients for power transmission (efficiency) and reflection and loss are respectively η<sub>P</sub>≡S<sub>21</sub><sup>2</sup>=S<sub>−2</sub><sup>2</sup>/S<sub>+1</sub><sup>2 </sup>and S<sub>11</sub><sup>2</sup>=S<sub>−1</sub><sup>2</sup>/S<sub>+1</sub><sup>2 </sup>and 1−S<sub>21</sub><sup>2</sup>−S<sub>11</sub><sup>2</sup>=(2Γ<sub>1</sub>A<sub>1</sub><sup>2</sup>+2Γ<sub>2</sub>A<sub>2</sub><sup>2</sup>)/S<sub>+1</sub><sup>2</sup>.
In some implementations, the parameters D<sub>1,2</sub>, U<sub>1,2 </sub>can be designed (engineered), since one can adjust the resonant frequencies ω<sub>1,2 </sub>(compared to the desired operating frequency ω) and the generator/load supply/drain rates κ<sub>1,2</sub>. Their choice can target the optimization of some system performancecharacteristic of interest.
In some examples, a goal can be to maximize the power transmission (efficiency) η<sub>P</sub>≡S<sub>21</sub><sup>2 </sup>of the system, so one would require
<FORM>η<sub>P</sub>′(D<sub>1,2</sub>)=η<sub>P</sub>′(U<sub>1,2</sub>)=0 (10)</FORM>
Since S<sub>21 </sub>(from Eq. (8)) is symmetric upon interchanging 1<img id="CUSTOMCHARACTER00004" he="1.78mm" wi="3.89mm" file="US20100148589A120100617P00003.TIF" imgcontent="character" imgformat="tif"/>2, the optimal values for D<sub>1,2 </sub>(determined by Eqs. (10)) will be equal, namely D<sub>1</sub>=D<sub>2</sub>≡D<sub>o</sub>, and similarly U<sub>1</sub>=U<sub>2</sub>≡U<sub>o</sub>. Then,
<maths id="MATHUS00012" num="00012"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>S</mi><mn>21</mn></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>UU</mi><mi>o</mi></msub></mrow><mrow><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>U</mi><mi>o</mi></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>D</mi><mi>o</mi></msub></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mi>U</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>11</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
and from the condition η<sub>P</sub>′(D<sub>o</sub>)=0 we get that, for fixed values of U and U<sub>o</sub>, the efficiency can be maximized for the following values of the symmetric detuning
<maths id="MATHUS00013" num="00013"><math overflow="scroll"><mtable><mtr><mtd><mtable><mtr><mtd><mrow><mrow><msub><mi>D</mi><mi>o</mi></msub><mo>=</mo><mrow><mo>±</mo><msqrt><mrow><msup><mi>U</mi><mn>2</mn></msup><mo></mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>U</mi><mi>o</mi></msub></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mrow><mi>if</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><mi>U</mi></mrow><mo>></mo><mrow><mn>1</mn><mo>+</mo><msub><mi>U</mi><mi>o</mi></msub></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn><mo>,</mo></mrow></mtd><mtd><mrow><mrow><mrow><mi>if</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><mi>U</mi></mrow><mo>≤</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>U</mi><mi>o</mi></msub></mrow></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>12</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
which, in the case U>1+U<sub>O</sub>, can be rewritten for the two frequencies at which the efficiency peaks as
<maths id="MATHUS00014" num="00014"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>ω</mi><mo>±</mo></msub><mo>=</mo><mrow><mfrac><mrow><mrow><msub><mi>ω</mi><mn>1</mn></msub><mo></mo><msub><mi>Γ</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mi>ω</mi><mn>2</mn></msub><mo></mo><msub><mi>Γ</mi><mn>1</mn></msub></mrow></mrow><mrow><msub><mi>Γ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>Γ</mi><mn>2</mn></msub></mrow></mfrac><mo>±</mo><mrow><mfrac><mrow><mn>2</mn><mo></mo><msqrt><mrow><msub><mi>Γ</mi><mn>1</mn></msub><mo></mo><msub><mi>Γ</mi><mn>2</mn></msub></mrow></msqrt></mrow><mrow><msub><mi>Γ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>Γ</mi><mn>2</mn></msub></mrow></mfrac><mo></mo><msqrt><mrow><msup><mi>κ</mi><mn>2</mn></msup><mo></mo><mrow><mrow><mo>(</mo><mrow><msub><mi>Γ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>κ</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>(</mo><mrow><msub><mi>Γ</mi><mn>2</mn></msub><mo>+</mo><msub><mi>κ</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow></mrow></msqrt></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>13</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
whose splitting we denote as δ<sub>P</sub>≡ <o ostyle="single">ω</o>− <o ostyle="single">ω</o><sub>−</sub>. Note that, at exact resonance ω<sub>1</sub>=ω<sub>2</sub>, and for Γ<sub>1</sub>=Γ<sub>2</sub>≡θ<sub>O </sub>and κ<sub>1</sub>=κ<sub>2</sub>≡κ<sub>O</sub>, we get δ<sub>P</sub>=2√{square root over (κ<sup>2</sup>−(Γ<sub>o</sub>+κ<sub>o</sub>)<sup>2</sup>)}<δ<sub>E</sub>, namely the transmissionpeak splitting is smaller than the normalmode splitting. Then, by substituting D<sub>O </sub>into η<sub>P </sub>from Eq. (12), from the condition η<sub>P</sub>′ (U<sub>O</sub>)=0 we get that, for fixed value of U, the efficiency can be maximized for
<maths id="MATHUS00015" num="00015"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>U</mi><msup><mi>o</mi><mo>*</mo></msup></msub><mo>=</mo><mrow><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi>U</mi><mn>2</mn></msup></mrow></msqrt><mo></mo><mover><mo>⇒</mo><mrow><mi>Eq</mi><mo>.</mo><mstyle><mspace width="0.6em" height="0.6ex"/></mstyle><mo></mo><mrow><mo>(</mo><mn>12</mn><mo>)</mo></mrow></mrow></mover><mo></mo><msub><mi>D</mi><msup><mi>o</mi><mo>*</mo></msup></msub></mrow><mo>=</mo><mn>0</mn></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>14</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
which is known as ‘critical coupling’ condition, whereas for U<sub>O</sub><U<sub>O* </sub>the system is called ‘undercoupled’ and for U<sub>O</sub>>U<sub>O* </sub>it is called ‘overcoupled’. The dependence of the efficiency on the frequency detuning D<sub>O </sub>for different values of U<sub>O </sub>(including the ‘criticalcoupling’ condition) are shown in FIG. 2(a,b). The overall optimal power efficiency using Eqs. (14) is
<maths id="MATHUS00016" num="00016"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><msub><mi>η</mi><msup><mi>P</mi><mo>*</mo></msup></msub><mo>≡</mo><mrow><msub><mi>η</mi><mi>P</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mi>D</mi><msup><mi>o</mi><mo>*</mo></msup></msub><mo>,</mo><msub><mi>U</mi><msup><mi>o</mi><mo>*</mo></msup></msub></mrow><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mrow><msub><mi>U</mi><msup><mi>o</mi><mo>*</mo></msup></msub><mo></mo><mn>1</mn></mrow><mrow><msub><mi>U</mi><msup><mi>o</mi><mo>*</mo></msup></msub><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><msup><mrow><mo>(</mo><mfrac><mi>U</mi><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi>U</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>15</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
which is again only a function of the couplingtoloss ratio U=κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)} and tends to unity when U>>1, as depicted in FIG. 2(c).
In some examples, a goal can be to minimize the power reflection at the side of the generator S<sub>11</sub><sup>2 </sup>and the load S<sub>22</sub><sup>2</sup>, so one would then need
<FORM>S<sub>11,22</sub>=0<img id="CUSTOMCHARACTER00005" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>(1∓U<sub>1</sub>−iD<sub>1</sub>)(1±U<sub>2</sub>−iD<sub>2</sub>)+U<sup>2</sup>=0 (16)</FORM>
The equations above present ‘impedance matching’ conditions. Again, the set of these conditions is symmetric upon interchanging 1<img id="CUSTOMCHARACTER00006" he="1.78mm" wi="3.89mm" file="US20100148589A120100617P00003.TIF" imgcontent="character" imgformat="tif"/>2, so, by substituting D<sub>1</sub>=D<sub>2</sub>≡D<sub>o </sub>and U<sub>1</sub>=U<sub>2</sub>≡U<sub>O </sub>into Eqs. (16), we get
<FORM>(1−iD<sub>o</sub>)<sup>2</sup>−U<sub>o</sub><sup>2</sup>+U<sup>2</sup>=0, (17)</FORM>
from which we easily find that the values of D<sub>O </sub>and U<sub>O </sub>that cancel all reflections are again exactly those in Eqs. (14).
It can be seen that, the two goals and their associated sets of conditions (Eqs. (10) and Eqs. (16)) result in the same optimized values of the intrasource and intradevice parameters D<sub>1,2</sub>, U<sub>1,2</sub>. Note that for a lossless system this would be an immediate consequence of power conservation (Hermiticity of the scattering matrix), but this is not apparent for a lossy system.
Accordingly, for any temporal energytransfer scheme, once the parameters specific only to the source or to the device (such as their resonant frequencies and their excitation or loading rates respectively) have been optimally designed, the efficiency monotonically increases with the ratio of the sourcedevice couplingrate to their loss rates. Using the definition of a resonance quality factor Q=ω/2Γ and defining by analogy the coupling factor k≡1/Q<sub>κ</sub>≡2κ/√{square root over (ω<sub>1</sub>ω<sub>2</sub>)}, it is therefore exactly this ratio
<maths id="MATHUS00017" num="00017"><math overflow="scroll"><mtable><mtr><mtd><mrow><mi>U</mi><mo>=</mo><mrow><mfrac><mi>κ</mi><msqrt><mrow><msub><mi>Γ</mi><mn>1</mn></msub><mo></mo><msub><mi>Γ</mi><mn>2</mn></msub></mrow></msqrt></mfrac><mo>=</mo><mrow><mi>k</mi><mo></mo><msqrt><mrow><msub><mi>Q</mi><mn>1</mn></msub><mo></mo><msub><mi>Q</mi><mn>2</mn></msub></mrow></msqrt></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>18</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
that has been set as a figureofmerit for any system under consideration for wireless energytransfer, along with the distance over which this ratio can be achieved (clearly, U will be a decreasing function of distance). The operating regime U>1 is sometimes called ‘strongcoupling’ regime and is a sufficient condition for efficient energytransfer. In particular, for U>1 we get, from Eq. (15), η<sub>P*</sub>>17%, large enough for many practical applications. Note that in some applications, U>0.1 may be sufficient. In applications where it is impossible or impractical to run wires to supply power to a device, U<0.1 may be considered sufficient. One skilled in the art will recognize that the sufficient U is application and specification dependent. The figureofmerit U may be called the strongcoupling factor. We will further show how to design systems with a large strongcoupling factor.
To achieve a large strongcoupling factor U, in some examples, the energytransfer application preferably uses resonant modes of high quality factors Q, corresponding to low (i.e. slow) intrinsicloss rates Γ. This condition can be satisfied by designing resonant modes where all loss mechanisms, typically radiation and absorption, are sufficiently suppressed.
This suggests that the coupling be implemented using, not the lossy radiative farfield, which should rather be suppressed, but the evanescent (nonlossy) stationary nearfield. To implement an energytransfer scheme, usually more appropriate are finite objects, namely ones that are topologically surrounded everywhere by air, into where the near field extends to achieve the coupling. Objects of finite extent do not generally support electromagnetic states that are exponentially decaying in all directions in air away from the objects, since Maxwell's Equations in free space imply that k<sup>2</sup>=ω<sup>2</sup>/c<sup>2</sup>, where k is the wave vector, ω the angular frequency, and c the speed of light, because of which one can show that such finite objects cannot support states of infinite Q, rather there always is some amount of radiation. However, very longlived (socalled “highQ”) states can be found, whose tails display the needed exponential or exponentiallike decay away from the resonant object over long enough distances before they turn oscillatory (radiative). The limiting surface, where this change in the field behavior happens, is called the “radiation caustic”, and, for the wireless energytransfer scheme to be based on the near field rather than the far/radiation field, the distance between the coupled objects must be such that one lies within the radiation caustic of the other. One typical way of achieving a high radiationQ (Q<sub>rad</sub>) is to design subwavelength resonant objects. When the size of an object is much smaller than the wavelength of radiation in free space, its electromagnetic field couples to radiation very weakly. Since the extent of the nearfield into the area surrounding a finitesized resonant object is set typically by the wavelength, in some examples, resonant objects of subwavelength size have significantly longer evanescent fieldtails. In other words, the radiation caustic is pushed far away from the object, so the electromagnetic mode enters the radiative regime only with a small amplitude.
Moreover, most realistic materials exhibit some nonzero amount of absorption, which can be frequency dependent, and thus cannot support states of infinite Q, rather there always is some amount of absorption. However, very longlived (“highQ”) states can be found, where electromagnetic modal energy is only weakly dissipated. Some typical ways of achieving a high absorptionQ (Q<sub>abs</sub>) is to use materials which exhibit very small absorption at the resonant frequency and/or to shape the field to be localized more inside the least lossy materials.
Furthermore, to achieve a large strongcoupling factor U, in some examples, the energytransfer application may use systems that achieve a high coupling factor k, corresponding to strong (i.e. fast) coupling rate κ, over distances larger than the characteristic sizes of the objects.
Since finitesized subwavelength resonant objects can often be designed to have high Q, as was discussed above and will be seen in examples later on, such objects may typically be chosen for the resonant deviceobject. In these cases, the electromagnetic field is, in some examples, of a quasistatic nature and the distance, up to which sufficient coupling can be achieved, is dictated by the decaylaw of this quasistatic field.
Note that in some examples, the resonant sourceobject may be immobile and thus less restricted in its allowed geometry and size. It can be therefore chosen to be large enough that the nearfield extent is not limited by the wavelength, and can thus have nearly infinite radiationQ. Some objects of nearly infinite extent, such as dielectric waveguides, can support guided modes, whose evanescent tails are decaying exponentially in the direction away from the object, slowly if tuned close to cutoff, therefore a good coupling can also be achieved over distances quite a few times larger than a characteristic size of the source and/or deviceobject.
In the following, examples of systems suitable for energy transfer of the type described above are described. We will demonstrate how to compute the CMT parameters ω<sub>1,2</sub>, Q<sub>1,2 </sub>and k described above and how to choose or design these parameters for particular examples in order to produce a desirable figureofmerit U=κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)}=k√{square root over (Q<sub>1</sub>Q<sub>2</sub>)} at a desired distance D. In some examples, this figureofmerit is maximized when ω<sub>1,2 </sub>are tuned close to a particular angular frequency ω<sub>U</sub>.
In some examples, one or more of the resonant objects are selfresonant conducting coils. Referring to FIG. 3, a conducting wire of length, l, and crosssectional radius, a, is wound into a helical coil of radius, r, and height, h, (namely with N=√{square root over (l<sup>2</sup>−h<sup>2</sup>)}/2πr number of turns), surrounded by air. As described below, the wire has distributed inductance and distributed capacitance, and therefore it supports a resonant mode of angular frequency ω. The nature of the resonance lies in the periodic exchange of energy from the electric field within the capacitance of the coil, due to the charge distribution ρ(x) across it, to the magnetic field in free space, due to the current distribution j(x) in the wire. In particular, the charge conservation equation ∇·j=iωρ implies that: (i) this periodic exchange is accompanied by a π/2 phaseshift between the current and the charge density profiles, namely the energy W contained in the coil is at certain points in time completely due to the current and at other points in time completely due to the charge, and (ii) if ρ<sub>l</sub>(x) and I(x) are respectively the linear charge and current densities in the wire, where x runs along the wire,
<maths id="MATHUS00018" num="00018"><math overflow="scroll"><mrow><msub><mi>q</mi><mi>o</mi></msub><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo>∫</mo><mrow><mrow><mo></mo><mi>x</mi></mrow><mo></mo><mrow><mo></mo><mrow><msub><mi>ρ</mi><mi>l</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo></mo></mrow></mrow></mrow></mrow></mrow></math></maths>
is the maximum amount of positive charge accumulated in one side of the coil (where an equal amount of negative charge always also accumulates in the other side to make the system neutral) and I<sub>o</sub>=max {I(x)} is the maximum positive value of the linear current distribution, then I<sub>O</sub>=ωq<sub>o</sub>. Then, one can define an effective total inductance L and an effective total capacitance C of the coil through the amount of energy W inside its resonant mode:
<maths id="MATHUS00019" num="00019"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mrow><mi>W</mi><mo>≡</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msubsup><mi>I</mi><mi>o</mi><mn>2</mn></msubsup><mo></mo><mi>L</mi></mrow></mrow><mo>⇒</mo><mi>L</mi></mrow><mo>=</mo><mrow><mfrac><msub><mi>μ</mi><mi>o</mi></msub><mrow><mn>4</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msubsup><mi>I</mi><mi>o</mi><mn>2</mn></msubsup></mrow></mfrac><mo></mo><mrow><mo>∫</mo><mrow><mo>∫</mo><mrow><mrow><mo></mo><mi>x</mi></mrow><mo></mo><mrow><mo></mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo></mo><mfrac><mrow><mrow><mi>j</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>·</mo><mrow><mi>j</mi><mo></mo><mrow><mo>(</mo><msup><mi>x</mi><mi>′</mi></msup><mo>)</mo></mrow></mrow></mrow><mrow><mo></mo><mrow><mi>x</mi><mo></mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo></mo></mrow></mfrac></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>19</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mrow><mi>W</mi><mo>≡</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msubsup><mi>q</mi><mi>o</mi><mn>2</mn></msubsup><mo></mo><mfrac><mn>1</mn><mi>C</mi></mfrac></mrow></mrow><mo>⇒</mo><mfrac><mn>1</mn><mi>C</mi></mfrac></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>ɛ</mi><mi>o</mi></msub><mo></mo><msubsup><mi>q</mi><mi>o</mi><mn>2</mn></msubsup></mrow></mfrac><mo></mo><mrow><mo>∫</mo><mrow><mo>∫</mo><mrow><mrow><mo></mo><mi>x</mi></mrow><mo></mo><mrow><mo></mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo></mo><mfrac><mrow><mrow><mi>ρ</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mi>ρ</mi><mo></mo><mrow><mo>(</mo><msup><mi>x</mi><mi>′</mi></msup><mo>)</mo></mrow></mrow></mrow><mrow><mo></mo><mrow><mi>x</mi><mo></mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo></mo></mrow></mfrac></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>20</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where μ<sub>O </sub>and ∈<sub>O </sub>are the magnetic permeability and electric permittivity of free space.
With these definitions, the resonant angular frequency and the effective impedance can be given by the formulas ω=1/√{square root over (LC)} and Z=√{square root over (L/C)} respectively.
Losses in this resonant system consist of ohmic (material absorption) loss inside the wire and radiative loss into free space. One can again define a total absorption resistance R<sub>abs </sub>from the amount of power absorbed inside the wire and a total radiation resistance R<sub>rad </sub>from the amount of power radiated due to electric and magneticdipole radiation:
<maths id="MATHUS00020" num="00020"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>P</mi><mi>abs</mi></msub><mo>≡</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msubsup><mi>I</mi><mi>o</mi><mn>2</mn></msubsup><mo></mo><msub><mi>R</mi><mi>abs</mi></msub></mrow></mrow><mo>⇒</mo><mrow><msub><mi>R</mi><mi>abs</mi></msub><mo>≈</mo><mrow><msub><mi>ζ</mi><mi>c</mi></msub><mo></mo><mrow><mfrac><mi>l</mi><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>a</mi></mrow></mfrac><mo>·</mo><mfrac><msubsup><mi>I</mi><mi>rms</mi><mn>2</mn></msubsup><msubsup><mi>I</mi><mi>o</mi><mn>2</mn></msubsup></mfrac></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>21</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><msub><mi>P</mi><mi>rad</mi></msub><mo>≡</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msubsup><mi>I</mi><mi>o</mi><mn>2</mn></msubsup><mo></mo><msub><mi>R</mi><mi>rad</mi></msub></mrow></mrow><mo>⇒</mo><mrow><msub><mi>R</mi><mi>rad</mi></msub><mo>≈</mo><mrow><mfrac><msub><mi>ζ</mi><mi>o</mi></msub><mrow><mn>6</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>π</mi></mrow></mfrac><mo></mo><mrow><mo>[</mo><mrow><msup><mrow><mo>(</mo><mfrac><mrow><mi>ω</mi><mo></mo><mrow><mo></mo><mi>p</mi><mo></mo></mrow></mrow><mi>c</mi></mfrac><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mfrac><mrow><mi>ω</mi><mo></mo><msqrt><mrow><mo></mo><mi>m</mi><mo></mo></mrow></msqrt></mrow><mi>c</mi></mfrac><mo>)</mo></mrow><mn>4</mn></msup></mrow><mo>]</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>22</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where c=1/√{square root over (μ<sub>o</sub>∈<sub>o</sub>)} and ζ<sub>o</sub>=√{square root over (μ<sub>o</sub>/∈<sub>o</sub>)} are the light velocity and light impedance in free space, the impedance ζ<sub>c </sub>is ζ<sub>c</sub>=1/σζ=√{square root over (μ<sub>o</sub>ω/2σ)} with σ the conductivity of the conductor and ζ the skin depth at the frequency ω,
<maths id="MATHUS00021" num="00021"><math overflow="scroll"><mrow><mrow><msubsup><mi>I</mi><mi>rms</mi><mn>2</mn></msubsup><mo>=</mo><mrow><mfrac><mn>1</mn><mi>l</mi></mfrac><mo></mo><mrow><mo>∫</mo><mrow><mrow><mo></mo><mi>x</mi></mrow><mo></mo><msup><mrow><mo></mo><mrow><mi>I</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo></mo></mrow><mn>2</mn></msup></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></maths>
p=∫dx rρ<sub>l</sub>(x) is the electricdipole moment of the coil and
<maths id="MATHUS00022" num="00022"><math overflow="scroll"><mrow><mi>m</mi><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo>∫</mo><mrow><mrow><mo></mo><mi>x</mi></mrow><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>r</mi><mo>×</mo><mrow><mi>j</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></math></maths>
is the magneticdipole moment of the coil. For the radiation resistance formula Eq. (22), the assumption of operation in the quasistatic regime (h,r<<λ=2πc/ω) has been used, which is the desired regime of a subwavelength resonance. With these definitions, the absorption and radiation quality factors of the resonance may be given by Q<sub>abs</sub>=Z/R<sub>abs </sub>and Q<sub>rad</sub>=Z/R<sub>rad </sub>respectively.
From Eq. (19)(22) it follows that to determine the resonance parameters one simply needs to know the current distribution j in the resonant coil. Solving Maxwell's equations to rigorously find the current distribution of the resonant electromagnetic eigenmode of a conductingwire coil is more involved than, for example, of a standard LC circuit, and we can find no exact solutions in the literature for coils of finite length, making an exact solution difficult. One could in principle write down an elaborate transmissionlinelike model, and solve it by brute force. We instead present a model that is (as described below) in good agreement (˜5%) with experiment. Observing that the finite extent of the conductor forming each coil imposes the boundary condition that the current has to be zero at the ends of the coil, since no current can leave the wire, we assume that the resonant mode of each coil is well approximated by a sinusoidal current profile along the length of the conducting wire. We shall be interested in the lowest mode, so if we denote by x the coordinate along the conductor, such that it runs from −l/2 to +l/2, then the current amplitude profile would have the form I(x)=I<sub>o </sub>cos (πx/l), where we have assumed that the current does not vary significantly along the wire circumference for a particular x, a valid assumption provided a<<r. It immediately follows from the continuity equation for charge that the linear charge density profile should be of the form ρ<sub>l</sub>(x)=ρ<sub>o </sub>sin(πx/l), and thus q<sub>o</sub>=∫<sub>0</sub><sup>l/2</sup>dxρ<sub>o</sub>sin(πx/l)=ρ<sub>o</sub>l/π. Using these sinusoidal profiles we find the socalled “selfinductance” L<sub>s </sub>and “selfcapacitance” C<sub>s </sub>of the coil by computing numerically the integrals Eq. (19) and (20); the associated frequency and effective impedance are ω<sub>s </sub>and Z<sub>s </sub>respectively. The “selfresistances” R<sub>s </sub>are given analytically by Eq. (21) and (22) using
<maths id="MATHUS00023" num="00023"><math overflow="scroll"><mtable><mtr><mtd><mrow><msubsup><mi>I</mi><mi>rms</mi><mn>2</mn></msubsup><mo>=</mo><mi/><mo></mo><mrow><mfrac><mn>1</mn><mi>l</mi></mfrac><mo></mo><mrow><msubsup><mo>∫</mo><mrow><mrow><mo></mo><mi>l</mi></mrow><mo>/</mo><mn>2</mn></mrow><mrow><mi>l</mi><mo>/</mo><mn>2</mn></mrow></msubsup><mo></mo><mstyle><mspace width="0.2em" height="0.2ex"/></mstyle><mo></mo><mrow><mrow><mo></mo><mi>x</mi></mrow><mo></mo><msup><mrow><mo></mo><mrow><msub><mi>I</mi><mi>o</mi></msub><mo></mo><mrow><mi>cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mrow><mi>x</mi><mo>/</mo><mi>l</mi></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo></mrow><mn>2</mn></msup></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>=</mo><mi/><mo></mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msubsup><mi>I</mi><mi>o</mi><mn>2</mn></msubsup></mrow></mrow><mo>,</mo><mrow><mo></mo><mi>p</mi><mo></mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mi/><mo></mo><mrow><msub><mi>q</mi><mi>o</mi></msub><mo></mo><msqrt><mrow><msup><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mi>π</mi></mfrac><mo></mo><mi>h</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mrow><mfrac><mrow><mn>4</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>N</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mrow><mi>cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>N</mi></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mrow><mn>4</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>N</mi><mn>2</mn></msup></mrow><mo></mo><mn>1</mn></mrow><mo>)</mo></mrow><mo></mo><mi>π</mi></mrow></mfrac><mo></mo><mi>r</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><mi>and</mi></mrow></mrow></mtd></mtr></mtable></math></maths><maths id="MATHUS000232" num="00023.2"><math overflow="scroll"><mrow><mrow><mrow><mo></mo><mi>m</mi><mo></mo></mrow><mo>=</mo><mrow><msub><mi>I</mi><mi>o</mi></msub><mo></mo><msqrt><mrow><msup><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mi>π</mi></mfrac><mo></mo><mi>N</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>r</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mrow><mfrac><mtable><mtr><mtd><mrow><mrow><mrow><mi>cos</mi><mo></mo><mrow><mo>(</mo><mrow><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>N</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>12</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>N</mi><mn>2</mn></msup></mrow><mo></mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mi>sin</mi><mo></mo><mrow><mo>(</mo><mrow><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>N</mi></mrow><mo>)</mo></mrow></mrow><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mrow><mi>N</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mn>4</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>N</mi><mn>2</mn></msup></mrow><mo></mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mtd></mtr></mtable><mrow><mrow><mo>(</mo><mrow><mrow><mn>16</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>N</mi><mn>4</mn></msup></mrow><mo></mo><mrow><mn>8</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>N</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo></mo><mi>π</mi></mrow></mfrac><mo></mo><mi>hr</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow></mrow><mo>,</mo></mrow></math></maths>
and therefore the associated Q<sub>s </sub>factors can be calculated.
The results for two examples of resonant coils with subwavelength modes of λ<sub>s</sub>/r≧70 (i.e. those highly suitable for nearfield coupling and well within the quasistatic limit) are presented in Table 1. Numerical results are shown for the wavelength and absorption, radiation and total loss rates, for the two different cases of subwavelengthcoil resonant modes. Note that, for conducting material, copper (σ=5.998·10̂−7 S/m) was used. It can be seen that expected quality factors at microwave frequencies are Q<sub>s,abs</sub>≧1000 and Q<sub>s,rad</sub>≧5000.
<tables id="TABLEUS00001" num="00001"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="1" colwidth="98pt" align="left"/><colspec colname="2" colwidth="21pt" align="center"/><colspec colname="3" colwidth="28pt" align="center"/><colspec colname="4" colwidth="28pt" align="center"/><colspec colname="5" colwidth="21pt" align="center"/><colspec colname="6" colwidth="21pt" align="center"/><thead><row><entry namest="1" nameend="6" rowsep="1">TABLE 1</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row><row><entry>single coil</entry><entry>λ<sub>s</sub>/r</entry><entry>f (MHz)</entry><entry>Q<sub>s,rad</sub></entry><entry>Q<sub>s,abs</sub></entry><entry>Q<sub>s</sub></entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></thead><tbody valign="top"><row><entry/></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="1" colwidth="98pt" align="left"/><colspec colname="2" colwidth="21pt" align="char" char="."/><colspec colname="3" colwidth="28pt" align="center"/><colspec colname="4" colwidth="28pt" align="char" char="."/><colspec colname="5" colwidth="21pt" align="center"/><colspec colname="6" colwidth="21pt" align="center"/><tbody valign="top"><row><entry>r = 30 cm, h = 20 cm, a = 1 cm,</entry><entry>74.7</entry><entry>13.39</entry><entry>4164</entry><entry>8170</entry><entry>2758</entry></row><row><entry>N = 4</entry></row><row><entry>r = 10 cm, h = 3 cm, a = 2 mm,</entry><entry>140</entry><entry>21.38</entry><entry>43919</entry><entry>3968</entry><entry>3639</entry></row><row><entry>N = 6</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
Referring to FIG. 4, in some examples, energy is transferred between two selfresonant conductingwire coils. The electric and magnetic fields are used to couple the different resonant conductingwire coils at a distance D between their centers. Usually, the electric coupling highly dominates over the magnetic coupling in the system under consideration for coils with h>>2r and, oppositely, the magnetic coupling highly dominates over the electric coupling for coils with h<<2r. Defining the charge and current distributions of two coils 1,2 respectively as ρ<sub>1,2</sub>(x) and j<sub>1,2</sub>(x), total charges and peak currents respectively as q<sub>1,2 </sub>and I<sub>1,2</sub>, and capacitances and inductances respectively as C<sub>1,2 </sub>and L<sub>1,2</sub>, which are the analogs of ρ(x), j(x), q<sub>o</sub>, I<sub>o</sub>, C and L for the singlecoil case and are therefore well defined, we can define their mutual capacitance and inductance through the total energy:
<maths id="MATHUS00024" num="00024"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mrow><mrow><mi>W</mi><mo>≡</mo><mrow><msub><mi>W</mi><mn>1</mn></msub><mo>+</mo><msub><mi>W</mi><mn>2</mn></msub><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mrow><mo>(</mo><mrow><mrow><msubsup><mi>q</mi><mn>1</mn><mo>*</mo></msubsup><mo></mo><msub><mi>q</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msubsup><mi>q</mi><mn>2</mn><mo>*</mo></msubsup><mo></mo><msub><mi>q</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow><mo>/</mo><msub><mi>M</mi><mi>C</mi></msub></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><mo>(</mo><mrow><mrow><msubsup><mi>I</mi><mn>1</mn><mo>*</mo></msubsup><mo></mo><msub><mi>I</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msubsup><mi>I</mi><mn>2</mn><mo>*</mo></msubsup><mo></mo><msub><mi>I</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow><mo></mo><msub><mi>M</mi><mi>L</mi></msub></mrow></mrow></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo>⇒</mo><mrow><mn>1</mn><mo>/</mo><msub><mi>M</mi><mi>C</mi></msub></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><mn>4</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>ɛ</mi><mn>0</mn></msub><mo></mo><msub><mi>q</mi><mn>1</mn></msub><mo></mo><msub><mi>q</mi><mn>2</mn></msub></mrow></mfrac><mo></mo><mrow><mo>∫</mo><mrow><mo>∫</mo><mrow><mrow><mo></mo><mi>x</mi></mrow><mo></mo><mrow><mo></mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo></mo><mfrac><mrow><mrow><msub><mi>ρ</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><msub><mi>ρ</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><msup><mi>x</mi><mi>′</mi></msup><mo>)</mo></mrow></mrow></mrow><mrow><mo></mo><mrow><mi>x</mi><mo></mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo></mo></mrow></mfrac><mo></mo><mi>u</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo><mstyle><mtext></mtext></mstyle><mo></mo><mrow><msub><mi>M</mi><mi>L</mi></msub><mo>=</mo><mrow><mfrac><msub><mi>μ</mi><mi>o</mi></msub><mrow><mn>4</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>I</mi><mn>1</mn></msub><mo></mo><msub><mi>I</mi><mn>2</mn></msub></mrow></mfrac><mo></mo><mrow><mo>∫</mo><mrow><mo>∫</mo><mrow><mrow><mo></mo><mi>x</mi></mrow><mo></mo><mrow><mo></mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo></mo><mfrac><mrow><mrow><msub><mi>j</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>·</mo><mrow><msub><mi>j</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><msup><mi>x</mi><mi>′</mi></msup><mo>)</mo></mrow></mrow></mrow><mrow><mo></mo><mrow><mi>x</mi><mo></mo><msup><mi>x</mi><mi>′</mi></msup></mrow><mo></mo></mrow></mfrac><mo></mo><mi>u</mi></mrow></mrow></mrow></mrow></mrow><mo>,</mo><mstyle><mtext></mtext></mstyle><mo></mo><mi>where</mi></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo></mo><mrow><mrow><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><msubsup><mi>q</mi><mn>1</mn><mn>2</mn></msubsup><mo>/</mo><msub><mi>C</mi><mn>1</mn></msub></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msubsup><mi>I</mi><mn>1</mn><mn>2</mn></msubsup><mo></mo><msub><mi>L</mi><mn>1</mn></msub></mrow></mrow></mrow><mo>,</mo><mrow><msub><mi>W</mi><mn>2</mn></msub><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mrow><msubsup><mi>q</mi><mn>2</mn><mn>2</mn></msubsup><mo>/</mo><msub><mi>C</mi><mn>2</mn></msub></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msubsup><mi>I</mi><mn>2</mn><mn>2</mn></msubsup><mo></mo><msub><mi>L</mi><mn>2</mn></msub></mrow></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>23</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
and the retardation factor of u=exp (iωx−x′/c) inside the integral can been ignored in the quasistatic regime D<<λ of interest, where each coil is within the near field of the other. With this definition, the coupling factor is given by k=√{square root over (C<sub>1</sub>C<sub>2</sub>)}/M<sub>C</sub>+M<sub>L</sub>/√{square root over (L<sub>1</sub>L<sub>2</sub>)}.
Therefore, to calculate the coupling rate between two selfresonant coils, again the current profiles are needed and, by using again the assumed sinusoidal current profiles, we compute numerically from Eq. (23) the mutual capacitance M<sub>C,s </sub>and inductance M<sub>L,s </sub>between two selfresonant coils at a distance D between their centers, and thus k=1/Q<sub>κ</sub>, is also determined.
<tables id="TABLEUS00002" num="00002"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="offset" colwidth="14pt" align="left"/><colspec colname="1" colwidth="91pt" align="left"/><colspec colname="2" colwidth="14pt" align="center"/><colspec colname="3" colwidth="35pt" align="center"/><colspec colname="4" colwidth="28pt" align="center"/><colspec colname="5" colwidth="35pt" align="center"/><thead><row><entry/><entry namest="offset" nameend="5" rowsep="1">TABLE 2</entry></row><row><entry/><entry namest="offset" nameend="5" align="center" rowsep="1"/></row><row><entry/><entry/><entry/><entry/><entry>Q<sub>κ </sub>=</entry><entry/></row><row><entry/><entry>pair of coils</entry><entry>D/r</entry><entry>Q</entry><entry>1/k</entry><entry>U</entry></row><row><entry/><entry namest="offset" nameend="5" align="center" rowsep="1"/></row></thead><tbody valign="top"><row><entry/></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="offset" colwidth="14pt" align="left"/><colspec colname="1" colwidth="91pt" align="left"/><colspec colname="2" colwidth="14pt" align="char" char="."/><colspec colname="3" colwidth="35pt" align="center"/><colspec colname="4" colwidth="28pt" align="char" char="."/><colspec colname="5" colwidth="35pt" align="char" char="."/><tbody valign="top"><row><entry/><entry>r = 30 cm, h = 20 cm,</entry><entry>3</entry><entry>2758</entry><entry>38.9</entry><entry>70.9</entry></row><row><entry/><entry>a = 1 cm, N = 4</entry><entry>5</entry><entry>2758</entry><entry>139.4</entry><entry>19.8</entry></row><row><entry/><entry>λ/r ≈ 75</entry><entry>7</entry><entry>2758</entry><entry>333.0</entry><entry>8.3</entry></row><row><entry/><entry>Q<sub>s</sub><sup>abs </sup>≈ 8170, Q<sub>s</sub><sup>rad </sup>≈ 4164</entry><entry>10</entry><entry>2758</entry><entry>818.9</entry><entry>3.4</entry></row><row><entry/><entry>r = 10 cm, h = 3 cm,</entry><entry>3</entry><entry>3639</entry><entry>61.4</entry><entry>59.3</entry></row><row><entry/><entry>a = 2 mm, N = 6</entry><entry>5</entry><entry>3639</entry><entry>232.5</entry><entry>15.7</entry></row><row><entry/><entry>λ/r ≈ 140</entry><entry>7</entry><entry>3639</entry><entry>587.5</entry><entry>6.2</entry></row><row><entry/><entry>Q<sub>s</sub><sup>abs </sup>≈ 3968, Q<sub>s</sub><sup>rad </sup>≈ 43919</entry><entry>10</entry><entry>3639</entry><entry>1580</entry><entry>2.3</entry></row><row><entry/><entry namest="offset" nameend="5" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
Referring to Table 2, relevant parameters are shown for exemplary examples featuring pairs or identical self resonant coils. Numerical results are presented for the average wavelength and loss rates of the two normal modes (individual values not shown), and also the coupling rate and figureofmerit as a function of the coupling distance D, for the two cases of modes presented in Table 1. It can be seen that for medium distances D/r=103 the expected couplingtoloss ratios are in the range U˜270.
An experimental realization of an example of the above described system for wireless energy transfer consists of two selfresonant coils, one of which (the source coil) is coupled inductively to an oscillating circuit, and the second (the device coil) is coupled inductively to a resistive load, as shown schematically in FIG. 5. Referring to FIG. 5, A is a single copper loop of radius 25 cm that is part of the driving circuit, which outputs a sine wave with frequency 9.9 MHz. s and d are respectively the source and device coils referred to in the text. B is a loop of wire attached to the load (“lightbulb”). The various κ's represent direct couplings between the objects. The angle between coil d and the loop A is adjusted so that their direct coupling is zero, while coils s and d are aligned coaxially. The direct coupling between B and A and between B and s is negligible.
The parameters for the two identical helical coils built for the experimental validation of the power transfer scheme were h=20 cm, a=3 mm, r=30 cm and N=5.25. Both coils are made of copper. Due to imperfections in the construction, the spacing between loops of the helix is not uniform, and we have encapsulated the uncertainty about their uniformity by attributing a 10% (2 cm) uncertainty to h. The expected resonant frequency given these dimensions is f<sub>0</sub>=10.56±0.3 MHz, which is approximately 5% off from the measured resonance at around 9.90 MHz.
The theoretical Q for the loops is estimated to be ˜2500 (assuming perfect copper of resistivity ρ=1/σ=1.7×10<sup>−8 </sup>Ωm) but the measured value is 950±50. We believe the discrepancy is mostly due to the effect of the layer of poorly conducting copper oxide on the surface of the copper wire, to which the current is confined by the short skin depth (˜20 μm) at this frequency. We have therefore used the experimentally observed Q (and Γ<sub>1</sub>=Γ<sub>2</sub>=Γ=ω/(2Q) derived from it) in all subsequent computations.
The coupling coefficient κ can be found experimentally by placing the two selfresonant coils (finetuned, by slightly adjusting h, to the same resonant frequency when isolated) a distance D apart and measuring the splitting in the frequencies of the two resonant modes in the transmission spectrum. According to Eq. (13) derived by coupledmode theory, the splitting in the transmission spectrum should be δ<sub>P</sub>=2√{square root over (κ<sup>2</sup>ΓΓ<sup>2</sup>)}, when κ<sub>A,B </sub>are kept very small by keeping A and B at a relatively large distance. The comparison between experimental and theoretical results as a function of distance when the two the coils are aligned coaxially is shown in FIG. 6.
FIG. 7 shows a comparison of experimental and theoretical values for the strongcoupling factor U=κ/Γ as a function of the separation between the two coils. The theory values are obtained by using the theoretically obtained κ and the experimentally measured Γ. The shaded area represents the spread in the theoretical U due to the ˜5% uncertainty in Q. As noted above, the maximum theoretical efficiency depends only on the parameter U, which is plotted as a function of distance in FIG. 7. U is greater than 1 even for D=2.4m (eight times the radius of the coils), thus the system is in the stronglycoupled regime throughout the entire range of distances probed.
The powergenerator circuit was a standard Colpitts oscillator coupled inductively to the source coil by means of a single loop of copper wire 25 cm in radius (see FIG. 5). The load consisted of a previously calibrated lightbulb, and was attached to its own loop of insulated wire, which was in turn placed in proximity of the device coil and inductively coupled to it. Thus, by varying the distance between the lightbulb and the device coil, the parameter U<sub>B</sub>=κ<sub>B</sub>/Γ was adjusted so that it matched its optimal value, given theoretically by Eq. (14) as U<sub>B*</sub>=√{square root over (1+U<sup>2</sup>)}. Because of its inductive nature, the loop connected to the lightbulb added a small reactive component to κ<sub>B </sub>which was compensated for by slightly retuning the coil. The work extracted was determined by adjusting the power going into the Colpitts oscillator until the lightbulb at the load was at its full nominal brightness.
In order to isolate the efficiency of the transfer taking place specifically between the source coil and the load, we measured the current at the midpoint of each of the selfresonant coils with a currentprobe (which was not found to lower the Q of the coils noticeably.) This gave a measurement of the current parameters I<sub>1 </sub>and I<sub>2 </sub>defined above. The power dissipated in each coil was then computed from P<sub>1,2</sub>=ΓLI<sub>1,2</sub><sup>2</sup>, and the efficiency was directly obtained from η=P<sub>B</sub>/(P<sub>1</sub>+P<sub>2</sub>+P<sub>B</sub>) . To ensure that the experimental setup was well described by a twoobject coupledmode theory model, we positioned the device coil such that its direct coupling to the copper loop attached to the Colpitts oscillator was zero. The experimental results are shown in FIG. 8, along with the theoretical prediction for maximum efficiency, given by Eq. (15).
Using this example, we were able to transmit significant amounts of power using this setup from the source coil to the device coil, fully lighting up a 60 W lightbulb from distances more than 2 m away, for example. As an additional test, we also measured the total power going into the driving circuit. The efficiency of the wireless powertransmission itself was hard to estimate in this way, however, as the efficiency of the Colpitts oscillator itself is not precisely known, although it is expected to be far from 100%. Nevertheless, this gave an overly conservative lower bound on the efficiency. When transmitting 60 W to the load over a distance of 2 m, for example, the power flowing into the driving circuit was 400 W. This yields an overall walltoload efficiency of ˜15%, which is reasonable given the expected ˜40% efficiency for the wireless power transmission at that distance and the low efficiency of the driving circuit.
From the theoretical treatment above, we see that in typical examples it is important that the coils be on resonance for the power transmission to be practical. We found experimentally that the power transmitted to the load dropped sharply as one of the coils was detuned from resonance. For a fractional detuning Δf/f<sub>0 </sub>of a few times the inverse loaded Q, the induced current in the device coil was indistinguishable from noise.
The power transmission was not found to be visibly affected as humans and various everyday objects, such as metallic and wooden furniture, as well as electronic devices large and small, were placed between the two coils, even when they drastically obstructed the line of sight between source and device. External objects were found to have an effect only when they were closer than 10 cm from either one of the coils. While some materials (such as aluminum foil, styrofoam and humans) mostly just shifted the resonant frequency, which could in principle be easily corrected with a feedback circuit of the type described earlier, others (cardboard, wood, and PVC) lowered Q when placed closer than a few centimeters from the coil, thereby lowering the efficiency of the transfer.
This method of power transmission is believed safe for humans. When transmitting 60 W (more than enough to power a laptop computer) across 2 m, we estimated that the magnitude of the magnetic field generated is much weaker than the Earth's magnetic field for all distances except for less than about 1 cm away from the wires in the coil, an indication of the safety of the scheme even after longterm use. The power radiated for these parameters was ˜5 W, which is roughly an order of magnitude higher than cell phones but could be drastically reduced, as discussed below.
Although the two coils are currently of identical dimensions, it is possible to make the device coil small enough to fit into portable devices without decreasing the efficiency. One could, for instance, maintain the product of the characteristic sizes of the source and device coils constant.
These experiments demonstrated experimentally a system for power transmission over medium range distances, and found that the experimental results match theory well in multiple independent and mutually consistent tests.
The efficiency of the scheme and the distances covered can be improved by silverplating the coils, which may increase their Q, or by working with more elaborate geometries for the resonant objects. Nevertheless, the performance characteristics of the system presented here are already at levels where they could be useful in practical applications.
In some examples, one or more of the resonant objects are capacitivelyloaded conducting loops or coils. Referring to FIG. 9 a helical coil with N turns of conducting wire, as described above, is connected to a pair of conducting parallel plates of area A spaced by distance d via a dielectric material of relative permittivity E, and everything is surrounded by air (as shown, N=1 and h=0). The plates have a capacitance C<sub>p</sub>=∈<sub>O</sub>ΓA/d, which is added to the distributed capacitance of the coil and thus modifies its resonance. Note however, that the presence of the loading capacitor may modify the current distribution inside the wire and therefore the total effective inductance L and total effective capacitance C of the coil may be different respectively from L<sub>s </sub>and C<sub>s</sub>, which are calculated for a selfresonant coil of the same geometry using a sinusoidal current profile. Since some charge may accumulate at the plates of the external loading capacitor, the charge distribution ρ inside the wire may be reduced, so C<C<sub>s</sub>, and thus, from the charge conservation equation, the current distribution j may flatten out, so L>L<sub>s</sub>. The resonant frequency for this system may be ω=1/√{square root over ((C+C<sub>p</sub>))}<ω<sub>s</sub>=1/√{square root over (L<sub>s</sub>C<sub>s</sub>)}, and I(x)→I<sub>o </sub>cos (πx/l)<img id="CUSTOMCHARACTER00007" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>C→C<sub>s</sub><img id="CUSTOMCHARACTER00008" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>ω→ω<sub>s</sub>, as C<sub>p</sub>→0.
In general, the desired CMT parameters can be found for this system, but again a very complicated solution of Maxwell's Equations is required. Instead, we will analyze only a special case, where a reasonable guess for the current distribution can be made. When C<sub>p</sub>>>C<sub>s</sub>>C, then ω≈1/√{square root over (LC<sub>p</sub>)}<<ω<sub>s </sub>and Z≈√{square root over (L/C<sub>p</sub>)}<<Z<sub>s</sub>, while all the charge is on the plates of the loading capacitor and thus the current distribution is constant along the wire. This allows us now to compute numerically L from Eq. (19). In the case h=0 and N integer, the integral in Eq. (19) can actually be computed analytically, giving the formula L=λ<sub>O</sub>r[ln(8r/a)−2]N<sup>2</sup>. Explicit analytical formulas are again available for R from Eq. (21) and (22), since I<sub>rms</sub>=I<sub>o</sub>, p≈0 and m=I<sub>O</sub>Nπr<sup>2 </sup>(namely only the magneticdipole term is contributing to radiation), so we can determine also Q<sub>abs</sub>=ωL/R<sub>abs </sub>and Q<sub>rad</sub>=ωL/R<sub>rad</sub>. At the end of the calculations, the validity of the assumption of constant current profile is confirmed by checking that indeed the condition C<sub>p</sub>>>C<sub>s</sub><img id="CUSTOMCHARACTER00009" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00002.TIF" imgcontent="character" imgformat="tif"/>ω<<ω<sub>s </sub>is satisfied. To satisfy this condition, one could use a large external capacitance, however, this would usually shift the operational frequency lower than the optimal frequency, which we will determine shortly; instead, in typical examples, one often prefers coils with very small selfcapacitance C<sub>s </sub>to begin with, which usually holds, for the types of coils under consideration, when N=1, so that the selfcapacitance comes from the charge distribution across the single turn, which is almost always very small, or when N>1 and h>>2Na, so that the dominant selfcapacitance comes from the charge distribution across adjacent turns, which is small if the separation between adjacent turns is large.
The external loading capacitance C<sub>p </sub>provides the freedom to tune the resonant frequency (for example by tuning A or d). Then, for the particular simple case h=0, for which we have analytical formulas, the total Q=ωL/(R<sub>abs</sub>+R<sub>rad</sub>) becomes highest at the optimal frequency
<maths id="MATHUS00025" num="00025"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>ω</mi><mi>Q</mi></msub><mo>=</mo><msup><mrow><mo>[</mo><mrow><mfrac><msup><mi>c</mi><mn>4</mn></msup><mi>π</mi></mfrac><mo></mo><mrow><msqrt><mfrac><msub><mi>ɛ</mi><mi>o</mi></msub><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>σ</mi></mrow></mfrac></msqrt><mo>·</mo><mfrac><mn>1</mn><mrow><mi>a</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>Nr</mi><mn>3</mn></msup></mrow></mfrac></mrow></mrow><mo>]</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>7</mn></mrow></msup></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>24</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
reaching the value
<maths id="MATHUS00026" num="00026"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>Q</mi><mi>max</mi></msub><mo>=</mo><mrow><mfrac><mn>6</mn><mrow><mn>7</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>π</mi></mrow></mfrac><mo></mo><mrow><msup><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>π</mi><mn>2</mn></msup><mo></mo><msub><mi>η</mi><mi>o</mi></msub><mo></mo><mfrac><mrow><mi>σ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>a</mi><mn>2</mn></msup><mo></mo><msup><mi>N</mi><mn>2</mn></msup></mrow><mi>r</mi></mfrac></mrow><mo>)</mo></mrow><mrow><mn>3</mn><mo>/</mo><mn>7</mn></mrow></msup><mo>·</mo><mrow><mrow><mo>[</mo><mrow><mrow><mi>ln</mi><mo></mo><mrow><mo>(</mo><mfrac><mrow><mn>8</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>r</mi></mrow><mi>a</mi></mfrac><mo>)</mo></mrow></mrow><mo></mo><mn>2</mn></mrow><mo>]</mo></mrow><mo>.</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>25</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
At lower frequencies Q is dominated by ohmic loss and at higher frequencies by radiation. Note, however, that the formulas above are accurate as long as ω<sub>Q</sub><<ω<sub>s </sub>and, as explained above, this holds almost always when N=1, and is usually less accurate when N>1, since h=0 usually implies a large selfcapacitance. A coil with large h can be used, if the selfcapacitance needs to be reduced compared to the external capacitance, but then the formulas for L and ω<sub>Q</sub>, Q<sub>max </sub>are again less accurate. Similar qualitative behavior is expected, but a more complicated theoretical model is needed for making quantitative predictions in that case.
The results of the above analysis for two examples of subwavelength modes of λ/r≧70 (namely highly suitable for nearfield coupling and well within the quasistatic limit) of coils with N=1 and h=0 at the optimal frequency Eq. (24) are presented in Table 3. To confirm the validity of constantcurrent assumption and the resulting analytical formulas, modesolving calculations were also performed using another completely independent method: computational 3D finiteelement frequencydomain (FEFD) simulations (which solve Maxwell's Equations in frequency domain exactly apart for spatial discretization) were conducted, in which the boundaries of the conductor were modeled using a complex impedance ζ<sub>c</sub>=√{square root over (μ<sub>o</sub>ω/2σ)} boundary condition, valid as long as ζ<sub>c</sub>/ζ<sub>o</sub><<1 (<10<sup>−5 </sup>for copper in the microwave). Table 3 shows Numerical FEFD (and in parentheses analytical) results for the wavelength and absorption, radiation and total loss rates, for two different cases of subwavelengthloop resonant modes. Note that copper was used for the conducting material (σ=5.998™10<sup>7 </sup>S/m). Specific parameters of the plot in FIG. 4 are highlighted in bold in the table. The two methods (analytical and computational) are in good agreement and show that, in some examples, the optimal frequency is in the lowMHz microwave range and the expected quality factors are Q<sub>abs</sub>≧1000 and Q<sub>rad</sub>>10000.
<tables id="TABLEUS00003" num="00003"><table frame="none" colsep="0" rowsep="0" pgwide="1"><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="1" colwidth="70pt" align="left"/><colspec colname="2" colwidth="49pt" align="center"/><colspec colname="3" colwidth="49pt" align="center"/><colspec colname="4" colwidth="49pt" align="center"/><colspec colname="5" colwidth="42pt" align="center"/><colspec colname="6" colwidth="42pt" align="center"/><thead><row><entry namest="1" nameend="6" rowsep="1">TABLE 3</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row><row><entry>single coil</entry><entry>λ/r</entry><entry>f</entry><entry>Q<sub>rad</sub></entry><entry>Q<sub>abs</sub></entry><entry>Q</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></thead><tbody valign="top"><row><entry>r = 30 cm, a = 2 cm</entry><entry>111.4 (112.4)</entry><entry>8.976 (8.897)</entry><entry>29546 (30512)</entry><entry>4886 (5117)</entry><entry>4193 (4381)</entry></row><row><entry>ε = 10, A = 138 cm<sup>2</sup>,</entry></row><row><entry>d = 4 mm</entry></row><row><entry>r = 10 cm, a = 2 mm</entry><entry>69.7 (70.4)</entry><entry>43.04 (42.61)</entry><entry>10702 (10727)</entry><entry>1545 (1604)</entry><entry>1350 (1395)</entry></row><row><entry>ε = 10, A = 3.14 cm<sup>2</sup>,</entry></row><row><entry>d = 1 mm</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
Referring to FIG. 10, in some examples, energy is transferred between two capacitivelyloaded coils. For the rate of energy transfer between two capacitivelyloaded coils 1 and 2 at distance D between their centers, the mutual inductance M<sub>L </sub>can be evaluated numerically from Eq. (23) by using constant current distributions in the case ω<<ω<sub>s</sub>. In the case h=0, the coupling may be only magnetic and again we have an analytical formula, which, in the quasistatic limit r<<D<<λ and for the relative orientation shown in FIG. 10, is M<sub>L</sub>≈πμ<sub>O</sub>/2·(r<sub>1</sub>r<sub>2</sub>)<sup>2</sup>N<sub>1</sub>N<sub>2</sub>/D<sup>3</sup>, which means that k∝(√{square root over (r<sub>1</sub>r<sub>2</sub>)}/D)<sup>3 </sup>may be independent of the frequency ω and the number of turns N<sub>1</sub>, N<sub>2</sub>. Consequently, the resultant coupling figureofmerit of interest is
<maths id="MATHUS00027" num="00027"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mi>U</mi><mo>=</mo><mrow><mrow><mi>k</mi><mo></mo><msqrt><mrow><msub><mi>Q</mi><mn>1</mn></msub><mo></mo><msub><mi>Q</mi><mn>2</mn></msub></mrow></msqrt></mrow><mo>≈</mo><mrow><msup><mrow><mo>(</mo><mfrac><msqrt><mrow><msub><mi>r</mi><mn>1</mn></msub><mo></mo><msub><mi>r</mi><mn>2</mn></msub></mrow></msqrt><mi>D</mi></mfrac><mo>)</mo></mrow><mn>3</mn></msup><mo>·</mo><mfrac><mrow><msup><mi>π</mi><mn>2</mn></msup><mo></mo><msub><mi>η</mi><mi>o</mi></msub><mo></mo><mrow><mfrac><msqrt><mrow><msub><mi>r</mi><mn>1</mn></msub><mo></mo><msub><mi>r</mi><mn>2</mn></msub></mrow></msqrt><mi>λ</mi></mfrac><mo>·</mo><msub><mi>N</mi><mn>1</mn></msub></mrow><mo></mo><msub><mi>N</mi><mn>2</mn></msub></mrow><mrow><munder><mo>∏</mo><mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mo>,</mo><mn>2</mn></mrow></munder><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mrow><mrow><msqrt><mfrac><mrow><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>η</mi><mi>o</mi></msub></mrow><mrow><mi>λ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>σ</mi></mrow></mfrac></msqrt><mo>·</mo><mfrac><msub><mi>r</mi><mi>j</mi></msub><msub><mi>a</mi><mi>j</mi></msub></mfrac></mrow><mo></mo><msub><mi>N</mi><mi>j</mi></msub></mrow><mo>+</mo></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mn>8</mn><mn>3</mn></mfrac><mo></mo><msup><mi>π</mi><mn>5</mn></msup><mo></mo><msup><mrow><msub><mi>η</mi><mi>o</mi></msub><mo></mo><mrow><mo>(</mo><mfrac><msub><mi>r</mi><mi>j</mi></msub><mi>λ</mi></mfrac><mo>)</mo></mrow></mrow><mn>4</mn></msup><mo></mo><msubsup><mi>N</mi><mi>j</mi><mn>2</mn></msubsup></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>26</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
which again is more accurate for N<sub>1</sub>=N<sub>2</sub>=1.
From Eq. (26) it can be seen that the optimal frequency ω<sub>U </sub>where the figureofmerit is maximized to the value U<sub>max</sub>, is close to the frequency ω<sub>Q</sub><sub><sub2>1</sub2></sub><sub>Q</sub><sub><sub2>2 </sub2></sub>at which Q<sub>1</sub>Q<sub>2 </sub>is maximized, since k does not depend much on frequency (at least for the distances D<<λ, of interest for which the quasistatic approximation is still valid). Therefore, the optimal frequency ω<sub>U</sub>≈ω<sub>q</sub><sub><sub2>1</sub2></sub><sub>Q</sub><sub><sub2>2 </sub2></sub>be mostly independent of the distance D between the two coils and may lie between the two frequencies ω<sub>Q</sub><sub><sub2>1 </sub2></sub>and ω<sub>Q</sub><sub><sub2>2 </sub2></sub>at which the singlecoil Q<sub>1 </sub>and Q<sub>2 </sub>respectively peak. For same coils, this optimal frequency is given by Eq. (24) and then the strongcoupling factor from Eq. (26) becomes
<maths id="MATHUS00028" num="00028"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>U</mi><mi>max</mi></msub><mo>=</mo><mrow><msub><mi>kQ</mi><mi>max</mi></msub><mo>≈</mo><mrow><mrow><msup><mrow><mo>(</mo><mfrac><mi>r</mi><mi>D</mi></mfrac><mo>)</mo></mrow><mn>3</mn></msup><mo>·</mo><mfrac><mn>3</mn><mn>7</mn></mfrac></mrow><mo></mo><mrow><msup><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>π</mi><mn>2</mn></msup><mo></mo><msub><mi>η</mi><mi>o</mi></msub><mo></mo><mfrac><mrow><mi>σ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>a</mi><mn>2</mn></msup><mo></mo><msup><mi>N</mi><mn>2</mn></msup></mrow><mi>r</mi></mfrac></mrow><mo>)</mo></mrow><mrow><mn>3</mn><mo>/</mo><mn>7</mn></mrow></msup><mo>.</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>27</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
In some examples, one can tune the capacitivelyloaded conducting loops or coils, so that their angular resonant frequencies are close to ω<sub>U </sub>within Γ<sub>U</sub>, which is half the angular frequency width for which U>U<sub>max</sub>/2.
Referring to Table 4, numerical FEFD and, in parentheses, analytical results based on the above are shown for two systems each composed of a matched pair of the loaded coils described in Table 3. The average wavelength and loss rates are shown along with the coupling rate and coupling to loss ratio figureofmerit U=κ/Γ as a function of the coupling distance D, for the two cases. Note that the average numerical Γ<sub>rad </sub>shown are slightly different from the singleloop value of FIG. 3. Analytical results for Γ<sub>rad </sub>are not shown but the singleloop value is used. (The specific parameters corresponding to the plot in FIG. 10 are highlighted with bold in the table.) Again we chose N=1 to make the constantcurrent assumption a good one and computed M<sub>L </sub>numerically from Eq. (23). Indeed the accuracy can be confirmed by their agreement with the computational FEFD modesolver simulations, which give κ through the frequency splitting of the two normal modes of the combined system (δ<sub>E</sub>=2κ from Eq. (4)). The results show that for medium distances D/r=103 the expected couplingtoloss ratios are in the range U˜0.5−50.
<tables id="TABLEUS00004" num="00004"><table frame="none" colsep="0" rowsep="0" pgwide="1"><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="1" colwidth="98pt" align="left"/><colspec colname="2" colwidth="14pt" align="center"/><colspec colname="3" colwidth="28pt" align="center"/><colspec colname="4" colwidth="35pt" align="center"/><colspec colname="5" colwidth="42pt" align="center"/><colspec colname="6" colwidth="42pt" align="center"/><thead><row><entry namest="1" nameend="6" rowsep="1">TABLE 4</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row><row><entry>pair of coils</entry><entry>D/r</entry><entry>Q<sup>rad</sup></entry><entry>Q = ω/2Γ</entry><entry>Q<sub>κ </sub>= ω/2κ</entry><entry>κ/Γ</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></thead><tbody valign="top"><row><entry/></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="1" colwidth="98pt" align="left"/><colspec colname="2" colwidth="14pt" align="char" char="."/><colspec colname="3" colwidth="28pt" align="center"/><colspec colname="4" colwidth="35pt" align="center"/><colspec colname="5" colwidth="42pt" align="center"/><colspec colname="6" colwidth="42pt" align="center"/><tbody valign="top"><row><entry>r = 30 cm, a = 2 cm</entry><entry>3</entry><entry>30729</entry><entry>4216</entry><entry>62.6 (63.7)</entry><entry>67.4 (68.7)</entry></row><row><entry>s = 10, A = 138 cm<sup>2</sup>,</entry><entry>5</entry><entry>29577</entry><entry>4194</entry><entry>235 (248)</entry><entry>17.8 (17.6)</entry></row><row><entry>d = 4 mm</entry><entry>7</entry><entry>29128</entry><entry>4185</entry><entry>589 (646)</entry><entry>7.1 (6.8)</entry></row><row><entry>λ/r ≈ 112</entry></row><row><entry>Q<sup>abs </sup>≈ 4886</entry><entry>10</entry><entry>28833</entry><entry>4177</entry><entry>1539 (1828)</entry><entry>2.7 (2.4)</entry></row><row><entry>r = 10 cm, a = 2 mm</entry><entry>3</entry><entry>10955</entry><entry>1355</entry><entry>85.4 (91.3)</entry><entry>15.9 (15.3)</entry></row><row><entry>ε = 10, A = 3.14 cm<sup>2</sup>, d = 1 mm</entry><entry>5</entry><entry>10740</entry><entry>1351</entry><entry>313 (356)</entry><entry>4.32 (3.92)</entry></row><row><entry>λ/r ≈ 70</entry><entry>7</entry><entry>10759</entry><entry>1351</entry><entry>754 (925)</entry><entry>1.79 (1.51)</entry></row><row><entry>Q<sup>abs </sup>≈ 1646</entry><entry>10</entry><entry>10756</entry><entry>1351</entry><entry>1895 (2617)</entry><entry>0.71 (0.53)</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
Referring to FIG. 11, to rederive and express Eq. (15) in terms of the parameters which are more directly accessible from particular resonant objects, such as the capacitivelyloaded conducting loops, one can consider the following circuitmodel of the system, where the inductances L<sub>s</sub>, L<sub>d </sub>represent the source and device loops respectively, R<sub>s</sub>, R<sub>d </sub>their respective losses, and C, C<sub>d </sub>are the required corresponding capacitances to achieve for both resonance at frequency ω. A voltage generator V<sub>g </sub>is considered to be connected to the source and a load resistance R<sub>l </sub>to the device. The mutual inductance is denoted by M.
Then from the source circuit at resonance (ωL=1/ωC):
<maths id="MATHUS00029" num="00029"><math overflow="scroll"><mtable><mtr><mtd><mtable><mtr><mtd><mrow><msub><mi>V</mi><mi>g</mi></msub><mo>=</mo><mi/><mo></mo><mrow><mrow><mrow><msub><mi>I</mi><mi>s</mi></msub><mo></mo><msub><mi>R</mi><mi>s</mi></msub></mrow><mo></mo><mrow><mi>j</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>ω</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>MI</mi><mi>d</mi></msub></mrow></mrow><mo>⇒</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msubsup><mi>V</mi><mi>g</mi><mo>*</mo></msubsup><mo></mo><msub><mi>I</mi><mi>s</mi></msub></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>=</mo><mi/><mo></mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mrow><mo></mo><msub><mi>I</mi><mi>s</mi></msub><mo></mo></mrow><mn>2</mn></msup><mo></mo><msub><mi>R</mi><mi>s</mi></msub></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><mi>j</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>ω</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msubsup><mi>MI</mi><mi>d</mi><mo>*</mo></msubsup><mo></mo><msub><mi>I</mi><mi>s</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>28</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
and from the device circuit at resonance (ωL<sub>d</sub>=1/ωC<sub>d</sub>):
<FORM>0=I<sub>d</sub>(R<sub>d</sub>+R<sub>l</sub>)−jωMI<sub>s</sub><img id="CUSTOMCHARACTER00010" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>jωMI<sub>s</sub>=I<sub>d</sub>(R<sub>d</sub>+R<sub>l</sub>) (29)</FORM>
So by substituting Eq. (29) to Eq. (28) and taking the real part (for timeaveraged power) we get:
<maths id="MATHUS00030" num="00030"><math overflow="scroll"><mtable><mtr><mtd><mtable><mtr><mtd><mrow><msub><mi>P</mi><mi>g</mi></msub><mo>=</mo><mi/><mo></mo><mrow><mi>Re</mi><mo></mo><mrow><mo>{</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msubsup><mi>V</mi><mi>g</mi><mo>*</mo></msubsup><mo></mo><msub><mi>I</mi><mi>s</mi></msub></mrow><mo>}</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mi/><mo></mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mrow><mo></mo><msub><mi>I</mi><mi>s</mi></msub><mo></mo></mrow><mn>2</mn></msup><mo></mo><msub><mi>R</mi><mi>s</mi></msub></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mrow><mo></mo><msub><mi>I</mi><mi>d</mi></msub><mo></mo></mrow><mn>2</mn></msup><mo></mo><mrow><mo>(</mo><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>l</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>=</mo><mi/><mo></mo><mrow><msub><mi>P</mi><mi>s</mi></msub><mo>+</mo><msub><mi>P</mi><mi>d</mi></msub><mo>+</mo><msub><mi>P</mi><mi>l</mi></msub></mrow></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mtd><mtd><mrow><mo>(</mo><mn>30</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where we identified the power delivered by the generator P<sub>g</sub>=Re{V<sub>g</sub>*I<sub>s</sub>/2, the power lost inside the source P<sub>s</sub>=I<sub>s</sub><sup>2</sup>R<sub>s</sub>/2, the power lost inside the device P<sub>d</sub>=I<sub>d</sub><sup>2</sup>R<sub>d</sub>/2 and the power delivered to the load P<sub>l</sub>=I<sub>d</sub><sup>2</sup>R<sub>l</sub>/2. Then, the power transmission efficiency is:
<maths id="MATHUS00031" num="00031"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>η</mi><mi>P</mi></msub><mo>≡</mo><mfrac><msub><mi>P</mi><mi>l</mi></msub><msub><mi>P</mi><mi>g</mi></msub></mfrac></mrow><mo>=</mo><mi/><mo></mo><mrow><mfrac><msub><mi>R</mi><mi>l</mi></msub><mrow><mrow><msup><mrow><mo></mo><mfrac><msub><mi>I</mi><mi>s</mi></msub><msub><mi>I</mi><mi>d</mi></msub></mfrac><mo></mo></mrow><mn>2</mn></msup><mo></mo><msub><mi>R</mi><mi>s</mi></msub></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>l</mi></msub></mrow></mrow></mrow></mfrac><mo></mo><mover><mo>=</mo><mrow><mo>(</mo><mn>29</mn><mo>)</mo></mrow></mover><mo></mo><mrow><mfrac><msub><mi>R</mi><mi>l</mi></msub><mrow><mrow><mfrac><mrow><mo>(</mo><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msubsup><mi>R</mi><mi>l</mi><mn>2</mn></msubsup></mrow></mrow><mrow><mo>(</mo><mrow><mi>ω</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>M</mi><mn>2</mn></msup></mrow></mrow></mfrac><mo></mo><msub><mi>R</mi><mi>s</mi></msub></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>l</mi></msub></mrow></mrow></mrow></mfrac><mo>.</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>31</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
If we now choose the load impedance R<sub>l </sub>to optimize the efficiency by η<sub>P</sub>′(R<sub>l</sub>)=0, we get the optimal load impedance
<maths id="MATHUS00032" num="00032"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><msub><mi>R</mi><msup><mi>l</mi><mo>*</mo></msup></msub><msub><mi>R</mi><mi>d</mi></msub></mfrac><mo>=</mo><msqrt><mrow><mn>1</mn><mo>+</mo><mfrac><msup><mrow><mo>(</mo><mrow><mi>ω</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>M</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mrow><msub><mi>R</mi><mi>s</mi></msub><mo></mo><msub><mi>R</mi><mi>d</mi></msub></mrow></mfrac></mrow></msqrt></mrow></mtd><mtd><mrow><mo>(</mo><mn>32</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
and the maximum possible efficiency
<maths id="MATHUS00033" num="00033"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mi>η</mi><msup><mi>P</mi><mo>*</mo></msup></msub><mo>=</mo><mrow><mfrac><mrow><mrow><msub><mi>R</mi><msup><mi>l</mi><mo>*</mo></msup></msub><mo>/</mo><msub><mi>R</mi><mi>d</mi></msub></mrow><mo></mo><mn>1</mn></mrow><mrow><mrow><msub><mi>R</mi><msup><mi>l</mi><mo>*</mo></msup></msub><mo>/</mo><msub><mi>R</mi><mi>d</mi></msub></mrow><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mrow><msup><mrow><mo>[</mo><mfrac><mrow><mi>ω</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mrow><mi>M</mi><mo>/</mo><msqrt><mrow><msub><mi>R</mi><mi>s</mi></msub><mo></mo><msub><mi>R</mi><mi>d</mi></msub></mrow></msqrt></mrow></mrow><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo>(</mo><mrow><mi>ω</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mrow><mi>M</mi><mo>/</mo><msqrt><mrow><msub><mi>R</mi><mi>s</mi></msub><mo></mo><msub><mi>R</mi><mi>d</mi></msub></mrow></msqrt></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mo>]</mo></mrow><mn>2</mn></msup><mo>.</mo></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>33</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
To check now the correspondence with the CMT model, note that κ<sub>l</sub>=R<sub>l</sub>/2L<sub>d</sub>, Γ<sub>d</sub>=R<sub>d</sub>/2L<sub>d</sub>, Γ<sub>s</sub>=R<sub>s</sub>/2L<sub>s</sub>, and κ ωM/2√{square root over (L<sub>s</sub>L<sub>d</sub>)}, so then U<sub>l</sub>=κ<sub>l</sub>/Γ<sub>d</sub>=R<sub>l</sub>/R<sub>d </sub>and U=κ/√{square root over (Γ<sub>s</sub>Γ<sub>d</sub>)}=ωM/√{square root over (R<sub>s</sub>R<sub>d</sub>)}. Therefore, the condition Eq. (32) is identical to the condition Eq. (14) and the optimal efficiency Eq. (33) is identical to the general Eq. (15). Indeed, as the CMT analysis predicted, to get a large efficiency, we need to design a system that has a large strongcoupling factor U.
The results above can be used to increase or optimize the performance of a wireless energy transfer system, which employs capacitivelyloaded coils. For example, from the scaling of Eq. (27) with the different system parameters, one sees that to maximize the system figureofmerit U, in some examples, one can:
 Decrease the resistivity of the conducting material. This can be achieved, for example, by using good conductors (such as copper or silver) and/or lowering the temperature. At very low temperatures one could use also superconducting materials to achieve extremely good performance.
 Increase the wire radius a. In typical examples, this action can be limited by physical size considerations. The purpose of this action is mainly to reduce the resistive losses in the wire by increasing the crosssectional area through which the electric current is flowing, so one could alternatively use also a Litz wire, or ribbon, or any low ACresistance structure, instead of a circular wire.
 For fixed desired distance D of energy transfer, increase the radius of the loop r. In typical examples, this action can be limited by physical size considerations.
 For fixed desired distance vs. loopsize ratio D/r, decrease the radius of the loop r. In typical examples, this action can be limited by physical size considerations.
 Increase the number of turns N. (Even though Eq. (27) is expected to be less accurate for N>1, qualitatively it still provides a good indication that we expect an improvement in the couplingtoloss ratio with increased N.) In typical examples, this action can be limited by physical size and possible voltage considerations, as will be discussed in following paragraphs.
 Adjust the alignment and orientation between the two coils. The figureofmerit is optimized when both cylindrical coils have exactly the same axis of cylindrical symmetry (namely they are “facing” each other). In some examples, particular mutual coil angles and orientations that lead to zero mutual inductance (such as the orientation where the axes of the two coils are perpendicular and the centers of the two coils are on one of the two axes) should be avoided.
 Finally, note that the height of the coil h is another available design parameter, which can have an impact on the performance similar to that of its radius r, and thus the design rules can be similar.
The above analysis technique can be used to design systems with desired parameters. For example, as listed below, the above described techniques can be used to determine the cross sectional radius a of the wire used to design a system including two same singleturn loops with a given radius in order to achieve a specific performance in terms of U=κ/Γ at a given D/r between them, when the loop material is copper (σ=5.998·10<sup>7</sup>S/m):
<FORM>D/r=5, U≧10, r=30 cm<img id="CUSTOMCHARACTER00011" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>a≧9 mm</FORM>
<FORM>D/r=5, U≧10, r=5 cm<img id="CUSTOMCHARACTER00012" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>a≧3.7 mm</FORM>
<FORM>D/r=5, U≧20, r=30 cm<img id="CUSTOMCHARACTER00013" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>a≧20 mm</FORM>
<FORM>D/r=5, U≧20, r=5 cm<img id="CUSTOMCHARACTER00014" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>a≧8.3 mm</FORM>
<FORM>D/r=10, U≧1, r=30 cm<img id="CUSTOMCHARACTER00015" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>a≧7 mm</FORM>
<FORM>D/r=10, U≧1, r=5 cm<img id="CUSTOMCHARACTER00016" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>a≧2.8 mm</FORM>
<FORM>D/r=10, U≧3, r=30 cm<img id="CUSTOMCHARACTER00017" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>a≧25 mm</FORM>
<FORM>D/r=10, U≧3, r=5 cm<img id="CUSTOMCHARACTER00018" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>a≧10 mm</FORM>
Similar analysis can be done for the case of two dissimilar loops. For example, in some examples, the device under consideration may be identified specifically (e.g. a laptop or a cell phone), so the dimensions of the device object (r<sub>d</sub>, h<sub>d</sub>, a<sub>d</sub>, N<sub>d</sub>) may be restricted. However, in some such examples, the restrictions on the source object (r<sub>s</sub>, h<sub>s</sub>, a<sub>s</sub>, N<sub>s</sub>) may be much less, since the source can, for example, be placed under the floor or on the ceiling. In such cases, the desired distance between the source and device may be fixed; in other cases it may be variable. Listed below are examples (simplified to the case N<sub>s</sub>=N<sub>d</sub>=1 and h<sub>s</sub>=h<sub>d</sub>=0) of how one can vary the dimensions of the source object to achieve a desired system performance in terms of U<sub>sd</sub>=κ/√{square root over (Γ<sub>s</sub>Γ<sub>d</sub>)}, when the material is again copper (σ=5.998·10<sup>7</sup>S/m):
<FORM>D=1.5 m, U<sub>sd</sub>≧15, r<sub>d</sub>=30 cm, a<sub>d</sub>=6 mm<img id="CUSTOMCHARACTER00019" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.158 m, a<sub>s</sub>>5 mm</FORM>
<FORM>D=1.5 m, U<sub>sd</sub>≧30, r<sub>d</sub>=30 cm, a<sub>d</sub>=6 mm<img id="CUSTOMCHARACTER00020" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.15 m, a<sub>s</sub>>33 mm</FORM>
<FORM>D=1.5 m, U<sub>sd</sub>≧1, r<sub>d</sub>=5 cm, a<sub>d</sub>=4 mm<img id="CUSTOMCHARACTER00021" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.119 m, a<sub>s</sub>>7 mm</FORM>
<FORM>D=1.5 m, U<sub>sd</sub>≧2, r<sub>d</sub>=5 cm, a<sub>d</sub>=4 mm<img id="CUSTOMCHARACTER00022" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.119 m, a<sub>s</sub>>52 mm</FORM>
<FORM>D=2 m, U<sub>sd</sub>≧10, r<sub>d</sub>=30 cm, a<sub>d</sub>=6 mm=<img id="CUSTOMCHARACTER00023" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.518 m, a<sub>s</sub>>7 mm</FORM>
<FORM>D=2 m, U<sub>sd</sub>≧20, r<sub>d</sub>=30 cm, a<sub>d</sub>=6 mm=<img id="CUSTOMCHARACTER00024" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.514 m, a<sub>s</sub>>50 mm</FORM>
<FORM>D=2 m, U<sub>sd</sub>≧0.5, r<sub>d</sub>=5 cm, a<sub>d</sub>=4 mm=<img id="CUSTOMCHARACTER00025" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.491 m, a<sub>s</sub>>5 mm</FORM>
<FORM>D=2 m, U<sub>sd</sub>≧1, r<sub>d</sub>=5 cm, a<sub>d</sub>=4 mm=<img id="CUSTOMCHARACTER00026" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.491 m, a<sub>s</sub>>36 mm</FORM>
As described below, in some examples, the quality factor Q of the resonant objects may be limited from external perturbations and thus varying the coil parameters may not lead to significant improvements in Q. In such cases, one can opt to increase the strongcoupling factor U by increasing the coupling factor k. The coupling does not depend on the frequency and may weakly depend on the number of turns. Therefore, in some examples, one can:
 Increase the wire radii a<sub>1 </sub>and a<sub>2</sub>. In typical examples, this action can be limited by physical size considerations.
 For fixed desired distance D of energy transfer, increase the radii of the coils r<sub>1 </sub>and r<sub>2</sub>. In typical examples, this action can be limited by physical size considerations.
 For fixed desired distance vs. coilsizes ratio D/√{square root over (r<sub>1</sub>r<sub>2</sub>)}, only the weak (logarithmic) dependence of the inductance remains, which suggests that one should decrease the radii of the coils r<sub>1 </sub>and r<sub>2</sub>. In typical examples, this action can be limited by physical size considerations.
 Adjust the alignment and orientation between the two coils. In typical examples, the coupling is optimized when both cylindrical coils have exactly the same axis of cylindrical symmetry (namely they are “facing” each other). Particular mutual coil angles and orientations that lead to zero mutual inductance (such as the orientation where the axes of the two coils are perpendicular and the centers of the two coils are on one of the two axes) should obviously be avoided.
 Finally, note that the heights of the coils h<sub>1 </sub>and h<sub>2 </sub>are other available design parameters, which can have an impact to the coupling similar to that of their radii r<sub>1 </sub>and r<sub>2</sub>, and thus the design rules can be similar.
Further practical considerations apart from efficiency, e.g. physical size limitations, will be discussed in detail below.
In embodiments, the dimensions of the resonant objects may be determined by the particular application. For example, when the application is powering a laptop or a cellphone, the device resonant object cannot have dimensions that exceed those of the laptop or cellphone respectively. For a system of two loops of specified dimensions, in terms of loop radii r<sub>s,d </sub>and wire radii a<sub>s,d</sub>, the independent parameters left to adjust for the system optimization are: the number of turns N<sub>s,d</sub>, the frequency f, the powerload consumption rate κ<sub>l</sub>=R<sub>l</sub>/2L<sub>d </sub>and the powergenerator feeding rate κ<sub>g</sub>=R<sub>g</sub>/2L<sub>s</sub>, where R<sub>g </sub>is the internal (characteristic) impedance of the generator.
In general, in various examples, the dependent variable that one may want to increase or optimize may be the overall efficiency η. However, other important variables may need to be taken into consideration upon system design. For example, in examples featuring capacitivelyloaded coils, the designs can be constrained by, the currents flowing inside the wires I<sub>s,d </sub>and other components and the voltages across the capacitors V<sub>s,d</sub>. These limitations can be important because for ˜Watt power applications the values for these parameters can be too large for the wires or the capacitors respectively to handle. Furthermore, the total loaded (by the load) quality factor of the device Q<sub>d[l]</sub>=ω/2(Γ<sub>d</sub>Γ<sub>l</sub>)=ωL<sub>d</sub>/(R<sub>d</sub>+R<sub>l</sub>) and the total loaded (by the generator) quality factor of the source Q<sub>s[g]</sub>=ω/2(Γ<sub>s</sub>+Γ<sub>g</sub>)=ωL<sub>s</sub>/(R<sub>s</sub>+R<sub>g</sub>) are quantities that should be preferably small, because to match the source and device resonant frequencies to within their Q's, when those are very large, can be challenging experimentally and more sensitive to slight variations. Lastly, the radiated powers P<sub>s,rad </sub>and P<sub>d,rad </sub>may need to be minimized for concerns about farfield interference and safety, even though, in general, for a magnetic, nonradiative scheme they are already typically small. In the following, we examine then the effects of each one of the independent variables on the dependent ones.
We define a new variable wp to express the powerload consumption rate for some particular value of U through U<sub>l</sub>=κ<sub>l</sub>/Γ<sub>d</sub>=√{square root over (1+wp·U<sup>2</sup>)}. Then, in some examples, values which may impact the choice of this rate are: U<sub>l</sub>=1<img id="CUSTOMCHARACTER00027" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00002.TIF" imgcontent="character" imgformat="tif"/>wp=0 to minimize the required energy stored in the source (and therefore I<sub>s </sub>and V<sub>s</sub>), U<sub>l</sub>=√{square root over (1+U<sup>2</sup>)}1<img id="CUSTOMCHARACTER00028" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00002.TIF" imgcontent="character" imgformat="tif"/>wp=1 to maximize the efficiency, as seen earlier, or U<sub>l</sub>>><img id="CUSTOMCHARACTER00029" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00002.TIF" imgcontent="character" imgformat="tif"/>wp>>1 to decrease the required energy stored in the device (and therefore I<sub>d </sub>and V<sub>d</sub>) and to decrease or minimize Q<sub>d[l]</sub>. Similar is the impact of the choice of the powergenerator feeding rate U<sub>g</sub>=κ<sub>g</sub>/Γ<sub>s</sub>, with the roles of the source/device and generator/load reversed.
In some examples, increasing N<sub>s </sub>and N<sub>d </sub>may increase Q<sub>s </sub>and Q<sub>d</sub>, and thus U and the efficiency significantly. It also may decrease the currents I<sub>s </sub>and I<sub>d</sub>, because the inductance of the loops may increase, and thus the energy W<sub>s,d</sub>=L<sub>s,d</sub>I<sub>s,d</sub><sup>2</sup>/2 required for given output power P<sub>l </sub>can be achieved with smaller currents. However, in some examples, increasing N<sub>d </sub>and thus Q<sub>d </sub>can increase Q<sub>d[l]</sub>, P<sub>d,rad </sub>and the voltage across the device capacitance V<sub>d</sub>. Similar can be the impact of increasing N<sub>s </sub>on Q<sub>s[g]</sub>, P<sub>s,rad </sub>and V<sub>s</sub>. As a conclusion, in some examples, the number of turns N<sub>s </sub>and N<sub>d </sub>may be chosen large enough (for high efficiency) but such that they allow for reasonable voltages, loaded Q's and/or powers radiated.
With respect to the resonant frequency, again, there may be an optimal one for efficiency. Close to that optimal frequency Q<sub>d[l]</sub> and/or Q<sub>s[g]</sub> can be approximately maximum. In some examples, for lower frequencies the currents may typically get larger but the voltages and radiated powers may get smaller. In some examples, one may choose the resonant frequency to maximize any of a number of system parameters or performance specifications, such as efficiency.
One way to decide on an operating regime for the system may be based on a graphical method. Consider two loops of r<sub>s</sub>=25 cm, r<sub>d</sub>=15 cm, h=h<sub>d</sub>=0, a<sub>s</sub>=a<sub>d</sub>=3 mm and distance D=2m between them. In FIG. 12, we plot some of the above dependent variables (currents, voltages and radiated powers normalized to 1 Watt of output power) in terms of frequency f and N<sub>d</sub>, given some choice for wp and N<sub>s</sub>. FIG. 12 depicts the trend of system performance explained above. In FIG. 13, we make a contour plot of the dependent variables as functions of both frequency and wp but for both N<sub>s </sub>and N<sub>d </sub>fixed. For example, in embodiments, a reasonable choice of parameters for the system of two loops with the dimensions given above may be: N<sub>s</sub>=2, N<sub>d</sub>=6, f=10 MHz and wp=10, which gives the following performance characteristics: η=20.6%, Q<sub>d[l]</sub>=1264, I<sub>s</sub>=7.2 A, I<sub>d</sub>=1.4 A, V<sub>s</sub>=2.55 kV, V<sub>d</sub>=2.30 kV, P<sub>s,rad</sub>=0.15 W, P<sub>d,rad</sub>=0.006 W. Note that the results in FIGS. 12, 13 and the calculated performance characteristics are made using the analytical formulas provided above, so they are expected to be less accurate for large values of N<sub>s</sub>, N<sub>d</sub>, but still may give a good estimate of the scalings and the orders of magnitude.
Finally, in embodiments, one could additionally optimize for the source dimensions, if, for example, only the device dimensions are limited. Namely, one can add r<sub>s </sub>and a<sub>s </sub>in the set of independent variables and optimize with respect to these all the dependent variables of the system. In embodiments, such an optimization may lead to improved results.
A straight conducting rod of length 2h and crosssectional radius a has distributed capacitance and distributed inductance, and therefore can support a resonant mode of angular frequency ω. Using the same procedure as in the case of selfresonant coils, one can define an effective total inductance L and an effective total capacitance C of the rod through formulas Eqs. (19) and (20). With these definitions, the resonant angular frequency and the effective impedance may be given again by the common formulas ω=1/√{square root over (LC)} and Z=√{square root over (L/C)} If respectively. To calculate the total inductance and capacitance, one can assume again a sinusoidal current profile along the length of the conducting wire. When interested in the lowest mode, if we denote by x the coordinate along the conductor, such that it runs from −h to +h, then the current amplitude profile may have the form I(x)=I<sub>o </sub>cos (πx/2h), since it has to be zero at the open ends of the rod. This is the wellknown halfwavelength electric dipole resonant mode.
In some examples, one or more of the resonant objects may be inductivelyloaded conducting rods. Referring to FIG. 14, a straight conducting rod of length 2h and crosssectional radius a, as in the previous paragraph, is cut into two equal pieces of length h, which may be connected via a coil wrapped around a magnetic material of relative permeability μ, and everything is surrounded by air. The coil has an inductance L<sub>C</sub>, which is added to the distributed inductance of the rod and thus modifies its resonance. Note however, that the presence of the centerloading inductor may modify the current distribution inside the wire and therefore the total effective inductance L and total effective capacitance C of the rod may be different respectively from L<sub>s </sub>and C<sub>s</sub>, which are calculated for a selfresonant rod of the same total length using a sinusoidal current profile, as in the previous paragraph. Since some current may be running inside the coil of the external loading inductor, the current distribution j inside the rod may be reduced, so L<L<sub>s</sub>, and thus, from the charge conservation equation, the linear charge distribution ρ<sub>l </sub>may flatten out towards the center (being positive in one side of the rod and negative in the other side of the rod, changing abruptly through the inductor), so C>C<sub>s</sub>. The resonant frequency for this system may be ω=1/√{square root over ((L+L<sub>c</sub>C)}<ω<sub>s</sub>=1/√{square root over (L<sub>s</sub>C<sub>s</sub>)}, and I(x)→I<sub>o </sub>cos (πx/2h)<img id="CUSTOMCHARACTER00030" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>L→L<sub>s</sub><img id="CUSTOMCHARACTER00031" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>ω<sub>s</sub>, as L<sub>c</sub>→0.
In general, the desired CMT parameters can be found for this system, but again a very complicated solution of Maxwell's Equations is generally required. In a special case, a reasonable estimate for the current distribution can be made. When L<sub>c</sub>>>L<sub>s</sub>>L, then ω≈1/√{square root over (L<sub>c</sub>C)}<<ω<sub>s </sub>and Z≈√{square root over (L<sub>c</sub>/C)}>>Z<sub>s</sub>, while the current distribution is triangular along the rod (with maximum at the centerloading inductor and zero at the ends) and thus the charge distribution may be positive constant on one half of the rod and equally negative constant on the other side of the rod. This allows us to compute numerically C from Eq. (20). In this case, the integral in Eq. (20) can actually be computed analytically, giving the formula 1/C=1/(π∈<sub>o</sub>h)[ln(h/a)−1]. Explicit analytical formulas are again available for R from Eq. (21) and (22), since I<sub>rms</sub>=I<sub>o</sub>, p=q<sub>o</sub>h and m=0 (namely only the electricdipole term is contributing to radiation), so we can determine also Q<sub>abs</sub>=1/ωCR<sub>abs </sub>and Q<sub>rad</sub>=1/ωCR<sub>rad</sub>. At the end of the calculations, the validity of the assumption of triangular current profile may be confirmed by checking that indeed the condition L<sub>c</sub>>>L<sub>s</sub><img id="CUSTOMCHARACTER00032" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00002.TIF" imgcontent="character" imgformat="tif"/><<ωz<sub>s </sub>is satisfied. This condition may be relatively easily satisfied, since typically a conducting rod has very small selfinductance L<sub>s </sub>to begin with.
Another important loss factor in this case is the resistive loss inside the coil of the external loading inductor L<sub>c </sub>and it may depend on the particular design of the inductor. In some examples, the inductor may be made of a Brooks coil, which is the coil geometry which, for fixed wire length, may demonstrate the highest inductance and thus quality factor. The Brooks coil geometry has N<sub>Bc </sub>turns of conducting wire of crosssectional radius a<sub>Bc </sub>wrapped around a cylindrically symmetric coil former, which forms a coil with a square crosssection of side r<sub>Bc</sub>, where the inner side of the square is also at radius r (and thus the outer side of the square is at radius 2r<sub>Bc</sub>), therefore N<sub>Bc</sub>≈(r<sub>Bc</sub>/2a<sub>Bc</sub>)<sup>2</sup>. The inductance of the coil is then L<sub>c</sub>=2.0285 μ<sub>o</sub>r<sub>Bc</sub>N<sub>Bc</sub><sup>2</sup>≈2.0285 μ<sub>o</sub>r<sub>Bc</sub><sup>5</sup><sub>Bc</sub>/8a<sub>Bc</sub><sup>4 </sup>and its resistance
<maths id="MATHUS00034" num="00034"><math overflow="scroll"><mrow><mrow><msub><mi>R</mi><mi>c</mi></msub><mo>≈</mo><mrow><mfrac><mn>1</mn><mi>σ</mi></mfrac><mo></mo><mfrac><msub><mi>l</mi><mi>Bc</mi></msub><mrow><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msubsup><mi>a</mi><mi>Bc</mi><mn>2</mn></msubsup></mrow></mfrac><mo></mo><msqrt><mrow><mn>1</mn><mo>+</mo><mrow><mfrac><mrow><msub><mi>μ</mi><mi>o</mi></msub><mo></mo><mi>ω</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>σ</mi></mrow><mn>2</mn></mfrac><mo></mo><msup><mrow><mo>(</mo><mfrac><msub><mi>a</mi><mi>Bc</mi></msub><mn>2</mn></mfrac><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow></msqrt></mrow></mrow><mo>,</mo></mrow></math></maths>
where the total wire length is l<sub>Bc</sub>≈2π(3r<sub>Bc</sub>/2)N<sub>Bc</sub>≈3πr<sub>Bc</sub><sup>3</sup>/4a<sub>Bc</sub><sup>2 </sup>and we have used an approximate squareroot law for the transition of the resistance from the dc to the ac limit as the skin depth varies with frequency.
The external loading inductance L<sub>c </sub>provides the freedom to tune the resonant frequency. For example, for a Brooks coil with a fixed size r<sub>Bc</sub>, the resonant frequency can be reduced by increasing the number of turns N<sub>Bc </sub>by decreasing the wire crosssectional radius a<sub>Bc</sub>. Then the desired resonant angular frequency ω=1/√{square root over (L<sub>c</sub>C)} may be achieved for a<sub>Bc</sub>≈(2.0285 μ<sub>o</sub>r<sub>Bc</sub><sup>5</sup>ω<sup>2</sup>C)<sup>1/4 </sup>and the resulting coil quality factor may be Q<sub>c</sub>≈0.169μ<sub>o</sub>Υr<sub>Bc</sub><sup>2</sup>ω/√{square root over (1+ω<sup>2</sup>μ<sub>o</sub>σ√{square root over (2.0285μ<sub>o</sub>(r<sub>Bc</sub>/4)<sup>5</sup>C)})}. Then, for the particular simple case L<sub>c</sub>>>L<sub>s</sub>, for which we have analytical formulas, the total Q=1/ωC(R<sub>C</sub>+R<sub>abs</sub>+R<sub>rad</sub>) becomes highest at some optimal frequency ω<sub>Q</sub>, reaching the value Q<sub>max</sub>, that may be determined by the loadinginductor specific design. For example, for the Brookscoil procedure described above, at the optimal frequency Q<sub>max</sub>≈Q<sub>c</sub>≈0.8(μ<sub>o</sub>σ<sup>2</sup>r<sup>3</sup><sub>Bc</sub>/C<sup>1/4</sup>. At lower frequencies it is dominated by ohmic loss inside the inductor coil and at higher frequencies by radiation. Note, again, that the above formulas are accurate as long as ω<sub>Q</sub><<ω<sub>s </sub>and, as explained above, this may be easy to design for by using a large inductance.
The results of the above analysis for two examples, using Brooks coils, of subwavelength modes of λ/h≧200 (namely highly suitable for nearfield coupling and well within the quasistatic limit) at the optimal frequency ω<sub>Q </sub>are presented in Table 5.
Table 5 shows in parentheses (for similarity to previous tables) analytical results for the wavelength and absorption, radiation and total loss rates, for two different cases of subwavelengthrod resonant modes. Note that copper was used for the conducting material (σ=5.998·10<sup>7</sup>S/m). The results show that, in some examples, the optimal frequency may be in the lowMHz microwave range and the expected quality factors may be Q<sub>abs</sub>≧1000 and Q<sub>rad</sub>≧100000.
<tables id="TABLEUS00005" num="00005"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="1" colwidth="56pt" align="left"/><colspec colname="2" colwidth="28pt" align="center"/><colspec colname="3" colwidth="35pt" align="center"/><colspec colname="4" colwidth="42pt" align="center"/><colspec colname="5" colwidth="28pt" align="center"/><colspec colname="6" colwidth="28pt" align="center"/><thead><row><entry namest="1" nameend="6" rowsep="1">TABLE 5</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row><row><entry>single rod</entry><entry>λ/h</entry><entry>f (MHz)</entry><entry>Q<sub>rad</sub></entry><entry>Q<sub>abs</sub></entry><entry>Q</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></thead><tbody valign="top"><row><entry/></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="1" colwidth="56pt" align="left"/><colspec colname="2" colwidth="28pt" align="center"/><colspec colname="3" colwidth="35pt" align="char" char="."/><colspec colname="4" colwidth="42pt" align="center"/><colspec colname="5" colwidth="28pt" align="center"/><colspec colname="6" colwidth="28pt" align="center"/><tbody valign="top"><row><entry>h = 30 cm,</entry><entry>(403.8)</entry><entry>(2.477)</entry><entry>(2.72 * 10<sup>6</sup>)</entry><entry>(7400)</entry><entry>(7380)</entry></row><row><entry>a = 2 cm</entry></row><row><entry>μ = 1, r<sub>Bc </sub>= 2 cm,</entry></row><row><entry>a<sub>Bc </sub>= 0.88 mm,</entry></row><row><entry>N<sub>Bc </sub>= 129</entry></row><row><entry>h = 10 cm,</entry><entry>(214.2)</entry><entry>(14.010)</entry><entry>(6.92 * 10<sup>5</sup>)</entry><entry>(3908)</entry><entry>(3886)</entry></row><row><entry>a = 2 mm μ = 1,</entry></row><row><entry>r<sub>Bc </sub>= 5 mm,</entry></row><row><entry>a<sub>Bc </sub>= 0.25 mm,</entry></row><row><entry>N<sub>Bc </sub>= 103</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
In some examples, energy may be transferred between two inductivelyloaded rods. For the rate of energy transfer between two inductivelyloaded rods 1 and 2 at distance D between their centers, the mutual capacitance M<sub>C </sub>can be evaluated numerically from Eq. (23) by using triangular current distributions in the case ω<<ω<sub>s</sub>. In this case, the coupling may be only electric and again we have an analytical formula, which, in the quasistatic limit h<<D<<λ, and for the relative orientation such that the two rods are aligned on the same axis, is 1/M<sub>C</sub>≈½π∈<sub>O</sub>·(h<sub>1</sub>h<sub>2</sub>)<sup>2</sup>/D<sup>3</sup>, which means that k∝(√{square root over (h<sub>1</sub>h<sub>2</sub>)}/D)<sup>3 </sup>is independent of the frequency ω. One can then get the resultant strongcoupling factor U.
It can be seen that the frequency ω<sub>U</sub>, where the figureofmerit is maximized to the value U<sub>max</sub>, is close to the frequency ω<sub>Q</sub><sub><sub2>1</sub2></sub><sub>Q</sub><sub><sub2>2</sub2></sub>, where Q<sub>1</sub>Q<sub>2 </sub>is maximized, since k does not depend much on frequency (at least for the distances D<<λ of interest for which the quasistatic approximation is still valid). Therefore, the optimal frequency ω<sub>U</sub>≈ω<sub>Q</sub><sub><sub2>1</sub2></sub><sub>Q</sub><sub><sub2>2 </sub2></sub>may be mostly independent of the distance D between the two rods and may lie between the two frequencies ω<sub>Q</sub><sub><sub2>1 </sub2></sub>and ω<sub>Q</sub><sub><sub2>2 </sub2></sub>at which the singlerod Q<sub>1 </sub>and Q<sub>2 </sub>respectively peak. In some typical examples, one can tune the inductivelyloaded conducting rods, so that their angular eigenfrequencies are close to ω<sub>U </sub>within Γ<sub>U</sub>, which is half the angular frequency width for which U>U<sub>max</sub>/2 .
Referring to Table 6, in parentheses (for similarity to previous tables) analytical results based on the above are shown for two systems each composed of a matched pair of the loaded rods described in Table 5. The average wavelength and loss rates are shown along with the coupling rate and coupling to loss ratio figureofmerit U=κ/F as a function of the coupling distance D, for the two cases. Note that for Γ<sub>rad </sub>the singlerod value is used. Again we chose L<sub>C</sub>>>L<sub>S </sub>to make the triangularcurrent assumption a good one and computed M<sub>C </sub>numerically from Eq. (23). The results show that for medium distances D/h=103 the expected couplingtoloss ratios are in the range U˜0.5−100.
<tables id="TABLEUS00006" num="00006"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="4"><colspec colname="1" colwidth="105pt" align="left"/><colspec colname="2" colwidth="28pt" align="center"/><colspec colname="3" colwidth="42pt" align="center"/><colspec colname="4" colwidth="42pt" align="center"/><thead><row><entry namest="1" nameend="4" rowsep="1">TABLE 6</entry></row><row><entry namest="1" nameend="4" align="center" rowsep="1"/></row><row><entry>pair of rods</entry><entry>D/h</entry><entry>Q<sub>κ </sub>= 1/k</entry><entry>U</entry></row><row><entry namest="1" nameend="4" align="center" rowsep="1"/></row></thead><tbody valign="top"><row><entry/></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="4"><colspec colname="1" colwidth="105pt" align="left"/><colspec colname="2" colwidth="28pt" align="char" char="."/><colspec colname="3" colwidth="42pt" align="center"/><colspec colname="4" colwidth="42pt" align="char" char="."/><tbody valign="top"><row><entry>h = 30 cm, a = 2 cm</entry><entry>3</entry><entry> (70.3)</entry><entry>(105.0)</entry></row><row><entry>μ = 1, r<sub>Bc </sub>= 2 cm, a<sub>Bc </sub>= 0.88 mm,</entry><entry>5</entry><entry> (389)</entry><entry>(19.0)</entry></row><row><entry>N<sub>Bc </sub>= 129</entry></row><row><entry>λ/h ≈ 404</entry><entry>7</entry><entry>(1115)</entry><entry>(6.62)</entry></row><row><entry>Q ≈ 7380</entry><entry>10</entry><entry>(3321)</entry><entry>(2.22)</entry></row><row><entry>h = 10 cm, a = 2 mm</entry><entry>3</entry><entry> (120)</entry><entry>(32.4)</entry></row><row><entry>μ = 1, r<sub>Bc </sub>= 5 mm, a<sub>Bc </sub>= 0.25 mm,</entry><entry>5</entry><entry> (664)</entry><entry>(5.85)</entry></row><row><entry>N<sub>Bc </sub>= 103</entry></row><row><entry>λ/h ≈ 214</entry><entry>7</entry><entry>(1900)</entry><entry>(2.05)</entry></row><row><entry>Q ≈ 3886</entry><entry>10</entry><entry>(5656)</entry><entry>(0.69)</entry></row><row><entry namest="1" nameend="4" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
In some examples, one or more of the resonant objects are dielectric objects, such as disks. Consider a two dimensional dielectric disk object, as shown in FIG. 15(a), of radius r and relative permittivity s surrounded by air that supports highQ “whisperinggallery” resonant modes. The loss mechanisms for the energy stored inside such a resonant system are radiation into free space and absorption inside the disk material. HighQ<sub>rad </sub>and longtailed subwavelength resonances can be achieved when the dielectric permittivity ∈ is large and the azimuthal field variations are slow (namely of small principal number m). Material absorption is related to the material loss tangent: Q<sub>abs </sub>˜Re{∈}/Im{∈}. Modesolving calculations for this type of disk resonances were performed using two independent methods: numerically, 2D finitedifference frequencydomain (FDFD) simulations (which solve Maxwell's Equations in frequency domain exactly apart for spatial discretization) were conducted with a resolution of 30 pts/r analytically, standard separation of variables (SV) in polar coordinates was used.
<tables id="TABLEUS00007" num="00007"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="5"><colspec colname="1" colwidth="35pt" align="left"/><colspec colname="2" colwidth="49pt" align="center"/><colspec colname="3" colwidth="49pt" align="center"/><colspec colname="4" colwidth="42pt" align="center"/><colspec colname="5" colwidth="42pt" align="center"/><thead><row><entry namest="1" nameend="5" rowsep="1">TABLE 7</entry></row><row><entry namest="1" nameend="5" align="center" rowsep="1"/></row><row><entry>single disk</entry><entry>λ/r</entry><entry>Q<sup>abs</sup></entry><entry>Q<sup>rad</sup></entry><entry>Q</entry></row><row><entry namest="1" nameend="5" align="center" rowsep="1"/></row></thead><tbody valign="top"><row><entry>Re{ε} =</entry><entry>20.01 (20.00)</entry><entry>10103 (10075)</entry><entry>1988 (1992)</entry><entry>1661 (1663)</entry></row><row><entry>147.7,</entry></row><row><entry>m = 2</entry></row><row><entry>Re{ε} =</entry><entry>9.952 (9.950)</entry><entry>10098 (10087)</entry><entry>9078 (9168)</entry><entry>4780 (4802)</entry></row><row><entry>65.6,</entry></row><row><entry>m = 3</entry></row><row><entry namest="1" nameend="5" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
The results for two TEpolarized dielectricdisk subwavelength modes of λ/r≧10 are presented in Table 7. Table 7 shows numerical FDFD (and in parentheses analytical SV) results for the wavelength and absorption, radiation and total loss rates, for two different cases of subwavelengthdisk resonant modes. Note that diskmaterial losstangent Im{∈}/Re{∈}=10<sup>−4 </sup>was used. (The specific parameters corresponding to the plot in FIG. 15(a) are highlighted with bold in the table.) The two methods have excellent agreement and imply that for a properly designed resonant lowlossdielectric object values of Q<sub>rad</sub>≧2000 and Q<sub>abs</sub>˜10000 are achievable. Note that for the 3D case the computational complexity would be immensely increased, while the physics would not be significantly different. For example, a spherical object of ∈=147.7 has a whispering gallery mode with m=2, Q<sub>rad</sub>=13962, and λ/r=17.
The required values of ∈, shown in Table 7, might at first seem unrealistically large. However, not only are there in the microwave regime (appropriate for approximately meterrange coupling applications) many materials that have both reasonably high enough dielectric constants and low losses (e.g. Titania, Barium tetratitanate, Lithium tantalite etc.), but also ∈ could signify instead the effective index of other known subwavelength surfacewave systems, such as surface modes on surfaces of metallic materials or plasmonic (metallike, negative∈) materials or metallodielectric photonic crystals or plasmonodielectric photonic crystals.
To calculate now the achievable rate of energy transfer between two disks 1 and 2, as shown in FIG. 15(b) we place them at distance D between their centers. Numerically, the FDFD modesolver simulations give K through the frequency splitting of the normal modes of the combined system (δ<sub>E</sub>=2κ from Eq. (4)), which are even and odd superpositions of the initial singledisk modes; analytically, using the expressions for the separationofvariables eigenfields E<sub>1,2(r) </sub>CMT gives K through
<FORM>κ=ω<sub>1</sub>/2·∫d<sup>3</sup>r∈<sub>2</sub>(r)E<sub>2</sub>*(r)E<sub>1</sub>(r)/∫d<sup>3</sup>r∈(r)E<sub>1</sub>(r)<sup>2</sup>,</FORM>
where ∈<sub>j</sub>(r) and ∈(r) are the dielectric functions that describe only the disk j (minus the constant ∈<sub>o </sub>background) and the whole space respectively. Then, for medium distances D/r=103 and for nonradiative coupling such that D<2r<sub>c</sub>, where r<sub>c</sub>=mλ/2π is the radius of the radiation caustic, the two methods agree very well, and we finally find, as shown in Table 8, strongcoupling factors in the range U˜150. Thus, for the analyzed examples, the achieved figureofmerit values are large enough to be useful for typical applications, as discussed below.
<tables id="TABLEUS00008" num="00008"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="1" colwidth="56pt" align="left"/><colspec colname="2" colwidth="14pt" align="center"/><colspec colname="3" colwidth="21pt" align="center"/><colspec colname="4" colwidth="35pt" align="center"/><colspec colname="5" colwidth="49pt" align="center"/><colspec colname="6" colwidth="42pt" align="center"/><thead><row><entry namest="1" nameend="6" rowsep="1">TABLE 8</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row><row><entry>two disks</entry><entry>D/r</entry><entry>Q<sup>rad</sup></entry><entry>Q = ω/2Γ</entry><entry>ω/2κ</entry><entry>κ/Γ</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></thead><tbody valign="top"><row><entry/></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="1" colwidth="56pt" align="left"/><colspec colname="2" colwidth="14pt" align="char" char="."/><colspec colname="3" colwidth="21pt" align="center"/><colspec colname="4" colwidth="35pt" align="center"/><colspec colname="5" colwidth="49pt" align="center"/><colspec colname="6" colwidth="42pt" align="center"/><tbody valign="top"><row><entry>Re{ε} = 147.7,</entry><entry>3</entry><entry>2478</entry><entry>1989</entry><entry>46.9 (47.5)</entry><entry>42.4 (35.0)</entry></row><row><entry>m = 2</entry><entry>5</entry><entry>2411</entry><entry>1946</entry><entry>298.0 (298.0)</entry><entry>6.5 (5.6)</entry></row><row><entry>λ/r ≈ 20</entry><entry>7</entry><entry>2196</entry><entry>1804</entry><entry>769.7 (770.2)</entry><entry>2.3 (2.2)</entry></row><row><entry>Q<sup>abs </sup>≈ 10093</entry><entry>10</entry><entry>2017</entry><entry>1681</entry><entry>1714 (1601)</entry><entry>0.98 (1.04)</entry></row><row><entry>Re{ε} = 65.6,</entry><entry>3</entry><entry>7972</entry><entry>4455</entry><entry>144 (140)</entry><entry>30.9 (34.3)</entry></row><row><entry>m = 3</entry><entry>5</entry><entry>9240</entry><entry>4824</entry><entry>2242 (2083)</entry><entry>2.2 (2.3)</entry></row><row><entry>λ/r ≈ 10</entry><entry>7</entry><entry>9187</entry><entry>4810</entry><entry>7485 (7417)</entry><entry>0.64 (0.65)</entry></row><row><entry>Q<sup>abs </sup>≈ 10096</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
Note that even though particular examples are presented and analyzed above as examples of systems that use resonant electromagnetic coupling for wireless energy transfer, those of selfresonant conducting coils, capacitivelyloaded resonant conducting coils, inductivelyloaded resonant conducting rods and resonant dielectric disks, any system that supports an electromagnetic mode with its electromagnetic energy extending much further than its size can be used for transferring energy. For example, there can be many abstract geometries with distributed capacitances and inductances that support the desired kind of resonances. In some examples, the resonant structure can be a dielectric sphere. In any one of these geometries, one can choose certain parameters to increase and/or optimize U or, if the Q's are limited by external factors, to increase and/or optimize for k or, if other system performance parameters are of importance, to optimize those.
Illustrative ExampleIn one example, consider a case of wireless energy transfer between two identical resonant conducting loops, labeled by L<sub>1 </sub>and L<sub>2</sub>. The loops are capacitivelyloaded and couple inductively via their mutual inductance. Let r<sub>A </sub>denote the loops' radii, N<sub>A </sub>their numbers of turns, and b<sub>A </sub>the radii of the wires making the loops. We also denote by D<sub>12 </sub>the centertocenter separation between the loops.
Resonant objects of this type have two main loss mechanisms: ohmic absorption and farfield radiation. Using the same theoretical method as in previous sections, we find that for r<sub>A</sub>=7 cm, b<sub>A</sub>=6 mm, and N<sub>A</sub>=15 turns, the quality factors for absorption, radiation and total loss are respectively, Q<sub>A,abs</sub>=πf/Γ<sub>A,abs</sub>=3.19×10<sup>4</sup>, Q<sub>A,rad</sub>=πf/Γ<sub>A,rad</sub>=2.6×10<sup>8 </sup>and Q<sub>A</sub>=πf/Γ<sub>A</sub>=2.84×10<sup>4 </sup>at a resonant frequency f=1.8×10<sup>7 </sup>Hz (remember that L<sub>1 </sub>and L<sub>2 </sub>are identical and have the same properties). Γ<sub>A,abs</sub>, Γ<sub>A,rad </sub>are respectively the rates of absorptive and radiative loss of L<sub>1 </sub>and L<sub>2</sub>, and the rate of coupling between L<sub>1 </sub>and L<sub>2 </sub>is denoted by κ<sub>12</sub>.
When the loops are in fixed distinct parallel planes separated by D<sub>12</sub>=1.4 m and have their centers on an axis (C) perpendicular to their planes, as shown in FIG. 17a (Left), the coupling factor for inductive coupling (ignoring retardation effects) is k<sub>12</sub>≡κ<sub>12</sub>/πf=7.68×10<sup>−5</sup>, independent of time, and thus the strongcoupling factor is U<sub>12</sub>≡k<sub>12</sub>Q<sub>A</sub>=2.18. This configuration of parallel loops corresponds to the largest possible coupling rate κ<sub>12 </sub>at the particular separation D<sub>12</sub>.
We find that the energy transferred to L<sub>2 </sub>is maximum at time T<sub>*</sub>=κt<sub>*</sub>=tan<sup>−1</sup>(2.18)=1.14<img id="CUSTOMCHARACTER00033" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00001.TIF" imgcontent="character" imgformat="tif"/>t<sub>*</sub>=4775(1/f) from Eq. (5), and constitutes η<sub>E*</sub>=29% of the initial total energy from Eq. (6a), as shown in FIG. 17a (Right). The energies radiated and absorbed are respectively η<sub>rad,E</sub>(t<sub>*</sub>)=7.2% and η<sub>abs,E</sub>(t<sub>*</sub>)=58.1% of the initial total energy, with 5.8% of the energy remaining in L<sub>1</sub>.
We would like to be able to further increase the efficiency of energy transfer between these two resonant objects at their distance D<sub>12</sub>. In some examples, one can use an intermediate resonator between the source and device resonators, so that energy is transferred more efficiently from the source to the device resonator indirectly through the intermediate resonator.
FIG. 16 shows a schematic that generally describes one example of the invention, in which energy is transferred wirelessly among three resonant objects. Referring to FIG. 16, energy is transferred, over a distance D<sub>1B</sub>, from a resonant source object R<sub>1 </sub>of characteristic size r<sub>l </sub>to a resonant intermediate object R<sub>B </sub>of characteristic size r<sub>B</sub>, and then, over an additional distance D<sub>B2</sub>, from the resonant intermediate object R<sub>B </sub>to a resonant device object R<sub>2 </sub>of characteristic size r<sub>2</sub>, where D<sub>1B</sub>+D<sub>B2</sub>=D. As described above, the source object R<sub>1 </sub>can be supplied power from a power generator that is coupled to the source object R<sub>1</sub>. In some examples, the power generator can be wirelessly, e.g., inductively, coupled to the source object R<sub>1</sub>. In some examples, the power generator can be connected to the source object R<sub>1 </sub>by means of a wire or cable. Also, the device object R<sub>2 </sub>can be connected to a power load that consumes energy transferred to the device object R<sub>2</sub>. For example, the device object can be connected to e.g. a resistor, a battery, or other device. All objects are resonant objects. The wireless nearfield energy transfer is performed using the field (e.g. the electromagnetic field or acoustic field) of the system of three resonant objects.
Different temporal schemes can be employed, depending on the application, to transfer energy among three resonant objects. Here we will consider a particularly simple but important scheme: a onetime finiteamount energytransfer scheme
Let again the source, intermediate and device objects be 1, B, 2 respectively and their resonance modes, which we will use for the energy exchange, have angular frequencies ω<sub>1,B,2</sub>, frequencywidths due to intrinsic (absorption, radiation etc.) losses Γ<sub>1,B,2 </sub>and (generally) vector fields Γ<sub>1,B,2 </sub>(r), normalized to unity energy. Once the three resonant objects are brought in proximity, they can interact and an appropriate analytical framework for modeling this resonant interaction is again that of the wellknown coupledmode theory (CMT), which can give a good description of the system for resonances having quality factors of at least, for example, 10. Then, using e<sup>−iωt </sup>time dependence, for the chain arrangement shown in FIG. 16, the field amplitudes can be shown to satisfy, to lowest order:
<maths id="MATHUS00035" num="00035"><math overflow="scroll"><mtable><mtr><mtd><mrow><mstyle><mspace width="4.4em" height="4.4ex"/></mstyle><mo></mo><mrow><mrow><mrow><mfrac><mo></mo><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo></mo><mrow><mi></mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>ω</mi><mn>1</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>11</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mrow><mn>1</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>B</mi></mrow></msub><mo></mo><mrow><msub><mi>a</mi><mi>B</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo></mo><mrow><mrow><mfrac><mo></mo><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo></mo><mrow><msub><mi>a</mi><mi>B</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo></mo><mrow><mi></mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>ω</mi><mi>B</mi></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mi>B</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><msub><mi>a</mi><mi>B</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mi>BB</mi></msub><mo></mo><mrow><msub><mi>a</mi><mi>B</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mrow><mi>B</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mn>1</mn></mrow></msub><mo></mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mrow><mi>B</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mn>2</mn></mrow></msub><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo></mo><mstyle><mspace width="4.4em" height="4.4ex"/></mstyle><mo></mo><mrow><mrow><mfrac><mo></mo><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo></mo><mrow><mi></mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>ω</mi><mn>2</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mn>2</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mn>22</mn></msub><mo></mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>B</mi></mrow></msub><mo></mo><mrow><msub><mi>a</mi><mi>B</mi></msub><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>34</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where κ<sub>11,BB,22 </sub>are the shifts in each object's frequency due to the presence of the other, which are a secondorder correction and can be absorbed into the eigenfrequencies by setting ω<sub>1,B,2</sub>→ω<sub>1,B,2</sub>+κ<sub>11,BB,22</sub>, and κ<sub>ij </sub>are the coupling coefficients, which from the reciprocity requirement of the system must satisfy κ<sub>ij</sub>=κ<sub>ij</sub>. Note that, in some examples, the direct coupling coefficient κ<sub>12 </sub>between the resonant objects 1 and 2 may be much smaller than the coupling coefficients κ<sub>1B </sub>and κ<sub>B2 </sub>between these two resonant objects with the intermediate object B, implying that the direct energy transfer between 1 and 2 is substantially dominated by the indirect energy transfer through the intermediate object. In some examples, if the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>are at least 5 times larger than the direct coupling rate κ<sub>12</sub>, using an intermediate resonator may lead to an improvement in terms of energy transfer efficiency, as the indirect transfer may dominate the direct transfer. Therefore, in the CMT Eqs. (34) above, the direct coupling coefficient κ<sub>12 </sub>has been ignored, to analyze those particular examples.
The three resonant modes of the combined system are found by substituting [a<sub>1</sub>(t) a<sub>B</sub>(t), a<sub>2</sub>(t)]=[A<sub>1</sub>, A<sub>B</sub>, A<sub>2</sub>]e<sup>−i <o ostyle="single">ω</o>t</sup>. When the resonators 1 and 2 are at exact resonance ω<sub>1</sub>=ω<sub>2</sub>=ω<sub>A </sub>and for Γ<sub>1</sub>=Γ<sub>2</sub>=Γ<sub>A</sub>, the resonant modes have complex resonant frequencies
<FORM><o ostyle="single">ω</o><sub>±</sub>=ω<sub>AB</sub>±√{square root over ((Δω<sub>AB</sub>)<sup>2</sup>+κ<sub>1B</sub><sup>2</sup>+κ<sub>B2</sub><sup>2</sup>)} and <o ostyle="single">ω</o><sub>ds</sub>=ω<sub>A</sub>−iΓ<sub>A</sub> (35a)</FORM>
where ω<sub>AB</sub>=[(ω<sub>A</sub>+ω<sub>B</sub>)−i(Γ<sub>A</sub>+Γ<sub>B</sub>)]/2, Δω<sub>AB</sub>(ω<sub>A</sub>−ω<sub>B</sub>)−i(Γ<sub>A</sub>−Γ<sub>B</sub>)/2 and whose splitting we denote as {tilde over (δ)}≡ <o ostyle="single">ω</o><sub>+</sub>− <o ostyle="single">ω</o><sub>−</sub>, and corresponding resonant field amplitudes
<maths id="MATHUS00036" num="00036"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mover><mi>V</mi><mo>></mo></mover><mo>±</mo></msub><mo>=</mo><mrow><msub><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>A</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mi>B</mi></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>±</mo></msub><mo>=</mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>κ</mi><mrow><mn>1</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>B</mi></mrow></msub></mtd></mtr><mtr><mtd><mrow><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>ω</mi><mi>AB</mi></msub></mrow><mo>∓</mo><msqrt><mrow><msup><mrow><mo>(</mo><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>ω</mi><mi>AB</mi></msub></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msubsup><mi>κ</mi><mrow><mn>1</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>B</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>κ</mi><mrow><mi>B</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></msqrt></mrow></mtd></mtr><mtr><mtd><msub><mi>κ</mi><mrow><mi>B</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mn>2</mn></mrow></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><mi>and</mi></mrow></mrow></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo></mo><mrow><msub><mover><mi>V</mi><mo>></mo></mover><mi>ds</mi></msub><mo>=</mo><mrow><msub><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>A</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mi>B</mi></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mi>ds</mi></msub><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mo></mo><msub><mi>κ</mi><mrow><mi>B</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>κ</mi><mrow><mn>1</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>B</mi></mrow></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mrow><mn>35</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
Note that, when all resonators are at exact resonance ω<sub>1</sub>=ω<sub>2</sub>(=ω<sub>A</sub>)=ω<sub>B </sub>and for Γ<sub>1</sub>=Γ<sub>2</sub>(=Γ<sub>A</sub>)=Γ<sub>B</sub>, we get Δω<sub>AB</sub>=0, {tilde over (δ)}<sub>E</sub>=2√{square root over (κ<sub>1B</sub><sup>2</sup>+κ<sub>B2</sub><sup>2</sup>)}, and then
<maths id="MATHUS00037" num="00037"><math overflow="scroll"><mtable><mtr><mtd><mrow><msub><mover><mi>ω</mi><mi>_</mi></mover><mo>±</mo></msub><mo>=</mo><mrow><mrow><mrow><msub><mi>ω</mi><mi>A</mi></msub><mo>±</mo><msqrt><mrow><msubsup><mi>κ</mi><mrow><mn>1</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>B</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>κ</mi><mrow><mi>B</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></msqrt></mrow><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mi>A</mi></msub><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><mi>and</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><msub><mover><mi>ω</mi><mi>_</mi></mover><mi>ds</mi></msub></mrow></mrow><mo>=</mo><mrow><msub><mi>ω</mi><mi>A</mi></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mi>A</mi></msub></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mrow><mn>36</mn><mo></mo><mi>a</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msub><mover><mi>V</mi><mo>></mo></mover><mo>±</mo></msub><mo>=</mo><mrow><msub><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>A</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mi>B</mi></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo>±</mo></msub><mo>=</mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>κ</mi><mrow><mn>1</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>B</mi></mrow></msub></mtd></mtr><mtr><mtd><mrow><mo>∓</mo><msqrt><mrow><msubsup><mi>κ</mi><mrow><mn>1</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>B</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>κ</mi><mrow><mi>B</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></msqrt></mrow></mtd></mtr><mtr><mtd><msub><mi>κ</mi><mrow><mi>B</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mn>2</mn></mrow></msub></mtd></mtr></mtable><mo>]</mo></mrow><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><mi>and</mi></mrow></mrow></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo></mo><mrow><mrow><msub><mover><mi>V</mi><mo>></mo></mover><mi>ds</mi></msub><mo>=</mo><mrow><msub><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>A</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mi>B</mi></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>2</mn></msub></mtd></mtr></mtable><mo>]</mo></mrow><mi>ds</mi></msub><mo>=</mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mo></mo><msub><mi>κ</mi><mrow><mi>B</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>κ</mi><mrow><mn>1</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>B</mi></mrow></msub></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mrow><mo>,</mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mrow><mn>36</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
namely we get that the resonant modes split to a lower frequency, a higher frequency and a same frequency mode.
Assume now that at time t=0 the source object 1 has finite energy a<sub>1</sub>(0)<sup>2</sup>, while the intermediate and device objects have a<sub>B</sub>(0)<sup>2</sup>=a<sub>2</sub>(0)<sup>2</sup>=0. Since the objects are coupled, energy will be transferred from 1 to B and from B to 2. With these initial conditions, Eqs. (34) can be solved, predicting the evolution of the fieldamplitudes. The energytransfer efficiency will be {tilde over (η)}<sub>E</sub>≡a<sub>2</sub>(t)<sup>2</sup>/a<sub>1</sub>(0)<sup>2</sup>. The ratio of energy converted to loss due to a specific loss mechanism in resonators 1, B and 2, with respective loss rates Γ<sub>1,loss</sub>, Γ<sub>B,loss </sub>and Γ<sub>2,loss </sub>will be {tilde over (η)}<sub>loss,E</sub>=∫<sub>o</sub><sup>t</sup>dτ[2Γ<sub>1,loss</sub>a<sub>1</sub>(τ)<sup>2</sup>+2τ<sub>B,loss</sub>a<sub>B</sub>(τ)<sup>2</sup>+2Γ<sub>2,loss</sub>a<sub>2</sub>(τ)<sup>2</sup>]/a<sub>1</sub>(0)<sup>2</sup>. At exact resonance ω<sub>1</sub>=ω<sub>2</sub>(=ω<sub>A</sub>)=ω<sub>B </sub>(an optimal condition) and in the special symmetric case Γ<sub>1</sub>=Γ<sub>2</sub>=Γ<sub>A </sub>and κ<sub>1B</sub>=κ<sub>B2</sub>=κ, the field amplitudes are
<maths id="MATHUS00038" num="00038"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><mrow><mo>(</mo><mover><mi>T</mi><mo>~</mo></mover><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi></mi><mrow><mrow><mo></mo><mi></mi></mrow><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>ω</mi><mi>A</mi></msub><mo></mo><mi>t</mi></mrow></msup><mo></mo><mrow><msup><mi></mi><mrow><mrow><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow><mo>/</mo><mover><mi>U</mi><mo>~</mo></mover></mrow></msup><mo></mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mrow><mover><mi>Δ</mi><mo>~</mo></mover><mo></mo><mfrac><mrow><mi>sin</mi><mo></mo><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></msqrt><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow><mo>)</mo></mrow></mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></msqrt></mfrac></mrow><mo>+</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mi>cos</mi><mo></mo><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></msqrt><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow><mo>)</mo></mrow></mrow><mo>+</mo><msup><mi></mi><mrow><mrow><mo></mo><mover><mi>Δ</mi><mo>~</mo></mover></mrow><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow></msup></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mrow><mn>37</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>a</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msub><mi>a</mi><mi>B</mi></msub><mo></mo><mrow><mo>(</mo><mover><mi>T</mi><mo>~</mo></mover><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi></mi><mrow><mrow><mo></mo><mi></mi></mrow><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>ω</mi><mi>A</mi></msub><mo></mo><mi>t</mi></mrow></msup><mo></mo><msup><mi></mi><mrow><mrow><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow><mo>/</mo><mover><mi>U</mi><mo>~</mo></mover></mrow></msup><mo></mo><mfrac><mrow><mi>sin</mi><mo></mo><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></msqrt><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow><mo>)</mo></mrow></mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></msqrt></mfrac></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mrow><mn>37</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><msub><mi>a</mi><mn>2</mn></msub><mo></mo><mrow><mo>(</mo><mover><mi>T</mi><mo>~</mo></mover><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msup><mi></mi><mrow><mrow><mo></mo><mi></mi></mrow><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>ω</mi><mi>A</mi></msub><mo></mo><mi>t</mi></mrow></msup><mo></mo><mrow><msup><mi></mi><mrow><mrow><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow><mo>/</mo><mover><mi>U</mi><mo>~</mo></mover></mrow></msup><mo></mo><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mrow><mover><mi>Δ</mi><mo>~</mo></mover><mo></mo><mfrac><mrow><mi>sin</mi><mo></mo><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></msqrt><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow><mo>)</mo></mrow></mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></msqrt></mfrac></mrow><mo>+</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mi>cos</mi><mo></mo><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></msqrt><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi></mi><mrow><mrow><mo></mo><mover><mi>Δ</mi><mo>~</mo></mover></mrow><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow></msup></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mrow><mn>37</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>c</mi></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where {tilde over (T)}≡√{square root over (2κt)}, {tilde over (Δ)}<sup>−1</sup>=2√2κ/(Γ<sub>B</sub>−Γ<sub>A</sub>) and Ũ=2/√{square root over (2)}κ/(Γ<sub>A</sub>+Γ<sub>B</sub>).
In some examples, the system designer can adjust the duration of the coupling t at will. In some examples, the duration t can be adjusted to maximize the device energy (and thus efficiency {tilde over (η)}<sub>E</sub>). Then, in the special case above, it can be inferred from Eq. (37c) that {tilde over (η)}<sub>E </sub>is maximized for the {tilde over (T)}={tilde over (T)}<sub>*</sub>, that satisfies
<maths id="MATHUS00039" num="00039"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mrow><mo>[</mo><mrow><mover><mi>Δ</mi><mo>~</mo></mover><mo></mo><mrow><mover><mi>U</mi><mo>~</mo></mover><mo></mo><mrow><mo>(</mo><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>]</mo></mrow><mo></mo><mfrac><mrow><mi>sin</mi><mo></mo><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></msqrt><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow><mo>)</mo></mrow></mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></msqrt></mfrac></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo></mo><mrow><mover><mi>Δ</mi><mo>~</mo></mover><mo></mo><mover><mrow><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>U</mi></mrow><mo>~</mo></mover></mrow></mrow><mo>)</mo></mrow><mo></mo><mrow><mo>[</mo><mrow><mrow><mi>cos</mi><mo></mo><mrow><mo>(</mo><mrow><msqrt><mrow><mn>1</mn><mo></mo><msup><mover><mi>Δ</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></msqrt><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow><mo>)</mo></mrow></mrow><mo></mo><msup><mi></mi><mrow><mover><mi>Δ</mi><mo>~</mo></mover><mo></mo><mover><mi>T</mi><mo>~</mo></mover></mrow></msup></mrow><mo>]</mo></mrow></mrow></mrow><mo>=</mo><mn>0.</mn></mrow></mtd><mtd><mrow><mo>(</mo><mn>38</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
In general, this equation may not have an obvious analytical solution, but it does admit a simple solution in the following two cases:
When Γ<sub>A</sub>=Γ<sub>B</sub><img id="CUSTOMCHARACTER00034" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00002.TIF" imgcontent="character" imgformat="tif"/>{tilde over (Δ)}=0, Ũ=√{square root over (2)}κ/Γ<sub>B</sub>, the energy transfer from resonator 1 to resonator 2 is maximized at
<FORM>{tilde over (T)}<sub>*</sub>({tilde over (Δ)}=0)=2 tan<sup>−1 </sup>Ũ (39)</FORM>
resulting in an energytransfer efficiency
<maths id="MATHUS00040" num="00040"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><msub><mover><mi>η</mi><mo>~</mo></mover><mi>E</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>~</mo></mover><mo>*</mo></msub><mo>,</mo><mrow><mover><mi>Δ</mi><mo>~</mo></mover><mo>=</mo><mn>0</mn></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><msup><mrow><mo>[</mo><mrow><mfrac><msup><mover><mi>U</mi><mo>~</mo></mover><mn>2</mn></msup><mrow><mn>1</mn><mo>+</mo><msup><mover><mi>U</mi><mo>~</mo></mover><mn>2</mn></msup></mrow></mfrac><mo></mo><mrow><mi>exp</mi><mo></mo><mrow><mo>(</mo><mrow><mo></mo><mfrac><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>tan</mi><mrow><mo></mo><mn>1</mn></mrow></msup><mo></mo><mover><mi>U</mi><mo>~</mo></mover></mrow><mover><mi>U</mi><mo>~</mo></mover></mfrac></mrow><mo>)</mo></mrow></mrow></mrow><mo>]</mo></mrow><mn>2</mn></msup></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>40</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
which has the form of the twoobject energytransfer efficiency of Eq. (6b) but squared. When Γ<sub>A</sub>=0<img id="CUSTOMCHARACTER00035" he="1.78mm" wi="3.13mm" file="US20100148589A120100617P00002.TIF" imgcontent="character" imgformat="tif"/>{tilde over (Δ)}<sup>−1</sup>=Ũ=2√{square root over (2)}κ/Γ<sub>B</sub>, the energy transfer from resonator 1 to resonator 2 is maximized at
<maths id="MATHUS00041" num="00041"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mover><mi>T</mi><mo>~</mo></mover><mo>*</mo></msub><mo></mo><mrow><mo>(</mo><mrow><msup><mover><mi>Δ</mi><mo>~</mo></mover><mrow><mo></mo><mn>1</mn></mrow></msup><mo>=</mo><mover><mi>U</mi><mo>~</mo></mover></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mrow><mi>π</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mrow><mover><mi>U</mi><mo>~</mo></mover><mo>/</mo><msqrt><mrow><msup><mover><mi>U</mi><mo>~</mo></mover><mn>2</mn></msup><mo></mo><mn>1</mn></mrow></msqrt></mrow></mrow><mo>,</mo><mrow><mover><mi>U</mi><mo>~</mo></mover><mo>></mo><mn>1</mn></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mi>∞</mi><mo>,</mo><mrow><mover><mi>U</mi><mo>~</mo></mover><mo>≤</mo><mn>1</mn></mrow></mrow></mtd></mtr></mtable></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>41</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
resulting in an energytransfer efficiency
<maths id="MATHUS00042" num="00042"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><msub><mi>η</mi><mi>E</mi></msub><mo></mo><mrow><mo>(</mo><mrow><msub><mover><mi>T</mi><mo>~</mo></mover><mo>*</mo></msub><mo>,</mo><mrow><msup><mover><mi>Δ</mi><mo>~</mo></mover><mrow><mo></mo><mn>1</mn></mrow></msup><mo>=</mo><mover><mi>U</mi><mo>~</mo></mover></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>·</mo><mrow><mo>{</mo><mtable><mtr><mtd><mrow><msup><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi>exp</mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mo></mo><mi>π</mi></mrow><mo>/</mo><msqrt><mrow><msup><mover><mi>U</mi><mo>~</mo></mover><mn>2</mn></msup><mo></mo><mn>1</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></mrow><mo>]</mo></mrow><mn>2</mn></msup><mo>,</mo><mrow><mover><mi>U</mi><mo>~</mo></mover><mo>></mo><mn>1</mn></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn><mo>,</mo><mrow><mover><mi>U</mi><mo>~</mo></mover><mo>≤</mo><mn>1.</mn></mrow></mrow></mtd></mtr></mtable></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>42</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
In both cases, and in general for any {tilde over (Δ)}, the efficiency is an increasing function of U. Therefore, once more resonators that have high quality factors are preferred. In some examples, one may design resonators with Q>50. In some examples, one may design resonators with Q>100.
Illustrative ExampleReturning to our illustrative example, in order to improve the ˜29% efficiency of the energy transfer from L<sub>1 </sub>to L<sub>2</sub>, while keeping the distance D<sub>12 </sub>separating them fixed, we propose to introduce an intermediate resonant object that couples strongly to both L<sub>1 </sub>and L<sub>2</sub>, while having the same resonant frequency as both of them. In one example, we take that mediator to also be a capacitivelyloaded conductingwire loop, which we label by L<sub>B</sub>. We place L<sub>B </sub>at equal distance (D<sub>1B</sub>=D<sub>B2</sub>=D<sub>12</sub>/2=0.7 m) from L<sub>1 </sub>and L<sub>2 </sub>such that its axis also lies on the same axis (C), and we orient it such that its plane is parallel to the planes of L<sub>1 </sub>and L<sub>2</sub>, which is the optimal orientation in terms of coupling. A schematic diagram of the threeloops configuration is depicted in FIG. 17b (Left).
In order for L<sub>B </sub>to couple strongly to L<sub>1 </sub>and L<sub>2</sub>, its size needs to be substantially larger than the size of L<sub>1 </sub>and L<sub>2</sub>. However, this increase in the size of L<sub>B </sub>has considerable drawback in the sense that it may also be accompanied by significant decrease in its radiation quality factor. This feature may often occur for the resonant systems of this type: stronger coupling can often be enabled by increasing the objects' size, but it may imply stronger radiation from the object in question. Large radiation may often be undesirable, because it could lead to farfield interference with other RF systems, and in some systems also because of safety concerns. For r<sub>B</sub>=70 cm, b<sub>B</sub>=1.5 cm, and N<sub>B</sub>=1 turn, we get Q<sub>B,rad</sub>=πf/Γ<sub>B,abs</sub>=7706, Q<sub>B,rad</sub>=πf/Γ<sub>B,rad</sub>=400, so Q<sub>B</sub>=πf/Γ<sub>B</sub>=380, and k<sub>1B</sub>≡κ<sub>1B</sub>/πf=k<sub>B2</sub>=0.0056, so Ũ=2√{square root over (2)}κ/(Γ<sub>A</sub>+Γ<sub>B</sub>)=5.94 and {tilde over (Δ)}<sup>−1</sup>=2√{square root over (2)}κ/(Γ<sub>B</sub>−Γ<sub>A</sub>)=6.1, at f=1.8×10<sup>7 </sup>Hz. Note that since the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>are ≈70 times larger than κ<sub>12</sub>, indeed we can ignore the direct coupling between L<sub>1 </sub>and L<sub>2</sub>, and focus only on the indirect energy transfer through the intermediate loop L<sub>B</sub>, therefore the analysis of the previous section can be used.
The optimum in energy transferred to L<sub>2 </sub>occurs at time {tilde over (T)}<sub>*</sub>=√{square root over (2)}κt<sub>*</sub>=3.21<img id="CUSTOMCHARACTER00036" he="2.79mm" wi="3.13mm" file="US20100148589A120100617P00004.TIF" imgcontent="character" imgformat="tif"/>t<sub>*</sub>=129 (1/f), calculated from Eq. (38), and is equal to {tilde over (η)}E*=61.5% of the initial energy from Eq. (37c). [Note that, since Q<sub>A</sub>>>Q<sub>B</sub>, we could have used the analytical conclusions of the case {tilde over (Δ)}<sup>−1</sup>=Ũ and then we would have gotten a very close approximation of {tilde over (T)}<sub>*</sub>=3.19 from Eq. (41).] The energy radiated is {tilde over (η)}<sub>rad,E</sub>(t<sub>*</sub>)=31.1%, while the energy absorbed is {tilde over (η)}<sub>abs,E</sub>(t<sub>*</sub>)=3.3%, and 4.1% of the initial energy is left in L<sub>1</sub>. Thus, the energy transferred, now indirectly, from L<sub>1 </sub>to L<sub>2 </sub>has increased by factor of 2 relative to the twoloops direct transfer case. Furthermore, the transfer time in the threeloops case is now ≈35 times shorter than in the twoloops direct transfer, because of the stronger coupling rate. The dynamics of the energy transfer in the threeloops case is shown in FIG. 17b (Right).
Note that the energy radiated in the threeloop system has undesirably increased by factor of 4 compared to the twoloop system. We would like to be able to achieve a similar improvement in energytransfer efficiency, while not allowing the radiated energy to increase. In this specification, we disclose that, in some examples, this can be achieved by appropriately varying the values of the coupling strengths between the three resonators.
4. Efficient EnergyTransfer by a Chain of Three Resonances with Adiabatically Varying Coupling Strengths
Consider again the system of a source resonator R<sub>1</sub>, a device resonator R<sub>2 </sub>and an intermediate resonator R<sub>B</sub>. For the purposes of the present analysis, R<sub>1 </sub>and R<sub>2 </sub>will be assumed to have negligible mutual interactions with each other, while each of them can be strongly coupled to R<sub>B</sub>, with coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>respectively. Note that, in some examples of wireless energy transfer systems, for the resonant object R<sub>B </sub>to be able to have strong coupling with other resonant objects, it may be accompanied with inferior loss properties compared to R<sub>1 </sub>and R<sub>2</sub>, usually in terms of substantially larger radiation losses.
It was seen in a previous section that, when the resonators 1 and 2 are at exact resonance ω<sub>1</sub>=ω<sub>2</sub>=ω<sub>A </sub>and for Γ<sub>1</sub>=Γ<sub>2</sub>=Γ<sub>A</sub>, the system supports a resonant state (eigenstate) with resonant frequency (eigenfrequency) <o ostyle="single">ω</o><sub>ds</sub>=ω<sub>A</sub>−iΓ<sub>A </sub>and resonant field amplitudes {right arrow over (V)}=−[κ<sub>B2 </sub>0 κ<sub>1B</sub>]<sup>T</sup>/√{square root over (κ<sub>1B</sub><sup>2</sup>κ<sub>B2</sub><sup>2</sup>)}, where here we normalized it. This eigenstate {right arrow over (V)}<sub>ds </sub>we call the “dark state” in analogy with atomic systems and the related phenomenon of Electromagnetically Induced Transparency (EIT), wherein complete population transfer between two quantum states through a third lossy state, coupled to each of the other two states, is enabled. The dark state is the most essential building block of our proposed efficient weaklyradiative energytransfer scheme, because it has no energy at all in the intermediate (lossy) resonator R<sub>B</sub>, i.e. a<sub>B</sub>(t)=0 ∀t whenever the threeobject system is in state {right arrow over (V)}<sub>ds</sub>. In fact, if Γ<sub>A</sub>→0, then this state is completely lossless, or if δ<sub>A,rad</sub>→0, then this state is completely nonradiative. Therefore, we disclose using predominantly this state to implement the wireless energy transfer. By doing that, the proposed energy transfer scheme can be made completely lossless, in the limit Γ<sub>A</sub>→0, no matter how large is the loss rate Γ<sub>B</sub>, as shown in FIG. 20, or completely nonradiative, in the limit Γ<sub>A,rad</sub>θ0, no matter how large is the radiative loss rate Γ<sub>B,rad</sub>. Since the energy transfer efficiency increases as the quality factors of the first (source) and second (device) resonances increase, one may design these resonators to have a high quality factor. In some examples, one may design resonators with Q<sub>A</sub>>50. In some examples, one may design resonators with Q<sub>A</sub>>100.
Let us demonstrate how it is possible to use the dark state for energy transfer with minimal loss. From the expression of {right arrow over (V)}<sub>ds </sub>one can see that, if the threeobject system is in the state {right arrow over (V)}<sub>ds</sub>, then, in general, there is energy in the source resonator with field amplitude proportional to the coupling rate κ<sub>B2 </sub>between the device resonator and the intermediate resonator, and there is also energy in the device resonator with field amplitude proportional to the coupling rate κ<sub>1B </sub>between the source resonator and the intermediate resonator. Then, κ<sub>1B</sub>=0 corresponds to all the system's energy being in R<sub>1</sub>, while κ<sub>B2</sub>=0 corresponds to all the system's energy being in R<sub>2</sub>.
So, the important considerations necessary to achieve efficient weaklyradiative energy transfer, consist of preparing the system initially in state {right arrow over (V)}<sub>ds </sub>and varying the coupling rates in time appropriately to evolve this state {right arrow over (V)}<sub>ds </sub>in a way that will cause energy transfer. Thus, if at t=0 all the energy is in R<sub>1</sub>, then one should have κ<sub>1B</sub>(t=0)<<κ<sub>B2</sub>(t=0), for example κ<sub>1B</sub>(t=0)=0 and κ<sub>B2</sub>(t=0)≠0. In order for the total energy of the system to end up in R<sub>2</sub>, at a time t<sub>EIT </sub>when the full variation of the coupling rates has been completed, we should have κ<sub>1B</sub>(t=t<sub>EIT</sub>)>>κ<sub>B2</sub>(t=t<sub>EIT</sub>), for example κ<sub>1B</sub>(t=t<sub>EIT</sub>)≠0 and κ<sub>B2</sub>(t=t<sub>EIT</sub>)=0. This ensures that the initial and final states of the threeobject system are parallel to {right arrow over (V)}<sub>ds</sub>. However, a second important consideration is to keep the threeobject system at all times in {right arrow over (V)}<sub>ds</sub>(t), even as κ<sub>1B</sub>(t) and κ<sub>B2</sub>(t) are varied in time. This is crucial in order to prevent the system's energy from getting into any of the two other eigenstates {right arrow over (V)}<sub>±</sub> and thus getting into the intermediate object R<sub>B</sub>, which may be highly radiative or lossy in general, as in the example of FIG. 17. This consideration requires changing κ<sub>1B</sub>(t) and κ<sub>B2</sub>(t) slowly enough so as to make the entire threeobject system adiabatically follow the time evolution of {right arrow over (V)}<sub>ds </sub>(t). The criterion for adiabatic following can be expressed, in analogy to the population transfer case as
<maths id="MATHUS00043" num="00043"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mo></mo><mrow><mo>〈</mo><mrow><msub><mover><mi>V</mi><mo>></mo></mover><mo>±</mo></msub><mo></mo><mfrac><mrow><mo></mo><msub><mover><mi>V</mi><mo>></mo></mover><mi>ds</mi></msub></mrow><mrow><mo></mo><mi>t</mi></mrow></mfrac></mrow><mo>〉</mo></mrow><mo></mo></mrow><mo></mo><mrow><mo></mo><mrow><msub><mover><mi>ω</mi><mi>_</mi></mover><mo>±</mo></msub><mo></mo><msub><mover><mi>ω</mi><mi>_</mi></mover><mi>ds</mi></msub></mrow><mo></mo></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>43</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where {right arrow over (V)}<sub>±</sub> are the remaining two eigenstates of the system, with corresponding eigenvalues <o ostyle="single">ω</o><sub>±</sub>, as shown earlier. Note that any functional dependence of the coupling rates with time (with duration parameter t<sub>EIT</sub>) will work, provided it satisfies the adiabaticity criterion Eq. (43) above. The time functional can be linear, sinusoidal (as in the illustrative example to follow) or the temporal analog of a Butterworth (maximally flat) taper, a Chebyshev taper, an exponential taper and the like.
Referring to FIG. 18, an example coupling rate adjustment system 300 for adjusting coupling rates for the one or more of the resonator structures R<sub>1</sub>, R<sub>2</sub>, or R<sub>B </sub>is shown. As described, the coupling rates between the first and intermediate resonator structures and the intermediate and second resonator structures are characterized by κ<sub>1B </sub>and κ<sub>B2 </sub>respectively. These coupling rates, κ<sub>1B </sub>and κ<sub>B2</sub>, are several times (e.g., 70 times) greater than the coupling rate κ<sub>12 </sub>between the first and second resonator structure. In some examples, the coupling rate adjustment system can be a mechanical, electrical, or electromechanical system for dynamically adjusting, e.g., rotating, or effecting a translational movement, of the one or more resonator structures with respect to each other.
In some examples, the coupling rate κ<sub>1B </sub>is much smaller than the coupling rate the coupling rate κ<sub>B2 </sub>at the beginning of the energy transfer. By the end, i.e., when a substantial amount of energy has been transferred from the first resonator structure R<sub>1 </sub>to the second resonator structure, R<sub>2</sub>, the coupling rate κ<sub>1B </sub>is much greater than the coupling rate κ<sub>B2</sub>. In some examples, the coupling rate κ<sub>1B </sub>can be set to a fixed value while the coupling rate κ<sub>B2 </sub>is being varied from its maximum to its minimum value. In some examples, the coupling rate κ<sub>B2 </sub>can be set to a fixed value while the coupling rate κ<sub>1B </sub>is being varied from its minimum to its maximum value. In some examples, the coupling rate κ<sub>1B </sub>can be varied from a minimum to a maximum value while the coupling rate κ<sub>B2 </sub>is being varied from its maximum to minimum value.
Referring now to FIG. 19, a graph for implementing an example coupling rate adjustment system 300 is shown. As shown, in some examples, the coupling rate κ<sub>1B </sub>is set at its minimum value at time, t=0, and increased as a function of time (see, for example, equation 44), while the coupling rate κ<sub>B2 </sub>is at its maximum value at t=0 and decreased as a function of time (see, for example, equation 45). Accordingly, at the beginning (t=0), the value of κ<sub>1B </sub>is much smaller than the value of κ<sub>B2</sub>. In some examples, the value of κ<sub>1B </sub>can be selected to be any value much smaller than the value of κ<sub>B2</sub>. During the wireless energy transfer, the value of κ<sub>1B </sub>is increased, while the value of κ<sub>B2 </sub>is decreased. After a predetermined amount of time t<sub>EIT </sub>has elapsed (e.g., after a substantial amount of energy has been transferred to the second resonator), the value of κ<sub>1B </sub>becomes much greater than the value of κ<sub>B2</sub>.
In some implementations, the coupling rate adjustment system 300 can effect an adjustment of coupling rates between the resonator structures by changing a relative orientation of one or more of the resonator structures with respect to each other. For example, referring again to FIG. 18, the first and second resonator structures, R<sub>1 </sub>and R<sub>2</sub>, can be rotated about their respective axes (e.g., varying angles θ<sub>1</sub>, and θ<sub>2</sub>), with respect to the intermediate resonator structure R<sub>B </sub>to simultaneously change κ<sub>1B </sub>and κ<sub>B2</sub>. Alternatively, the orientation of the intermediate resonator structure, R<sub>B</sub>, can be adjusted, e.g., rotated about an axis, with respect to the first and second resonator structures to simultaneously change κ<sub>1B </sub>and κ<sub>B2</sub>. Alternatively, the orientation of only the first resonator structure R<sub>1 </sub>can be rotated about its respective axis to change κ<sub>1B</sub>, while the orientations of R<sub>2 </sub>and R<sub>B </sub>are fixed and thus κ<sub>B2 </sub>is fixed to a value intermediate between the minimum and maximum values of κ<sub>1B</sub>. Alternatively, the orientation of only the second resonator structure R<sub>2 </sub>can be rotated about its respective axis to change κ<sub>B2</sub>, while the orientations of R<sub>1 </sub>and R<sub>B </sub>are fixed and thus κ<sub>1B </sub>is fixed to a value intermediate between the minimum and maximum values of κ<sub>B2</sub>.
In some implementations, the coupling rate adjustment system 300 can effect an adjustment of coupling rates between the resonator structures by translationally moving one or more of the resonator structures with respect to each other. For example, the positions of the first and second resonator structures, R<sub>1 </sub>and R<sub>2</sub>, can be adjusted, e.g., moved along an axis, with respect to the intermediate resonator structure R<sub>B </sub>to simultaneously change κ<sub>1B </sub>and κ<sub>B2</sub>. Alternatively, a position of the intermediate resonator structure, R<sub>B</sub>, can be adjusted, e.g., moved along an axis, with respect to the first and second resonator structures to simultaneously change κ<sub>1B </sub>and κ<sub>B2</sub>. Alternatively, a position of only the first resonator structure, R<sub>1</sub>, can be adjusted, e.g., moved along an axis, with respect to the intermediate R<sub>B </sub>resonator structure to change κ<sub>1B</sub>, while the positions of R<sub>2 </sub>and R<sub>B </sub>are fixed and thus κ<sub>B2 </sub>is fixed to a value intermediate between the minimum and maximum values of κ<sub>1B</sub>. Alternatively, a position of only the second resonator structure, R<sub>2</sub>, can be adjusted, e.g., moved along an axis, with respect to the intermediate R<sub>B </sub>resonator structure to change κ<sub>B2</sub>, while the positions of R<sub>1 </sub>and R<sub>B </sub>are fixed and thus κ<sub>1B </sub>is fixed to a value intermediate between the minimum and maximum values of κ<sub>B2</sub>.
In some examples, the coupling rate adjustment system 300 can dynamically adjust an effective size of the resonator objects to effect adjustments in the coupling rates κ<sub>1B </sub>and κ<sub>B2 </sub>similar to that described above. The effective size can be adjusted by changing a physical size of the resonator objects. For example, the physical size can be adjusted by effecting mechanical changes in area, length, or other physical aspect of one or more of the resonator structures. The effective size can also be adjusted through nonmechanical changes, such as, but not limited to, applying a magnetic field to change the permeability of the one or more of the resonator structures.
In principle, one would think of making the transfer time t<sub>EIT </sub>as long as possible to ensure adiabaticity. However there is limitation on how slow the transfer process can optimally be, imposed by the losses in R<sub>1 </sub>and R<sub>2</sub>. Such a limitation may not be a strong concern in typical atomic EIT case, because the initial and final states there can be chosen to be nonlossy ground states. However, in our case, losses in R<sub>1 </sub>and R<sub>2 </sub>are not avoidable, and can be detrimental to the energy transfer process whenever the transfer time t<sub>EIT </sub>is not less than 1/Γ<sub>A</sub>. This is because, even if the threeobject system is carefully kept in {right arrow over (V)}<sub>ds </sub>at all times, the total energy of the system will decrease from its initial value as a consequence of losses in R<sub>1 </sub>and R<sub>2</sub>. Thus the duration of the transfer may be a compromise between these two limits: the desire to keep t<sub>EIT </sub>long enough to ensure nearadiabaticity, but short enough not to suffer from losses in R<sub>1 </sub>and R<sub>2</sub>.
Given a particular functional variation of the coupling rates with time with variation duration parameter t<sub>EIT</sub>, one may calculate the optimal energy transfer efficiency in the following way: First, for each t<sub>EIT</sub>, determine the optimal time t<sub>*</sub>, at which the energy at R<sub>2 </sub>is maximized and the transfer process may be be terminated. Then find the optimal variation time t<sub>EIT* </sub>based on the counteracting mechanisms discussed above. The optimal efficiency of energy transfer {tilde over (η)}<sub>EIT,E* </sub>can then be calculated. In most cases, this procedure may need to be done numerically using the CMT Eqs. (34) as analytical solutions may not be possible. With respect to optimizing the functional dependence of the coupling rates with time, one may choose one that minimizes the coupling of energy to the eigenstates {right arrow over (V)}<sub>±</sub> for a given t<sub>EIT</sub>, which may lead to the temporal analog of a Chebyshev taper.
In some examples, the optimal t<sub>EIT </sub>may not be long enough for the adiabadicity criterion of Eq. (43) to be always satisfied. In those cases, some energy may get into at least one of the lossy states {right arrow over (V)}<sub>±</sub>. Still significant improvement in efficiency and radiation loss may be achieved by the mode of operation where the coupling rates are variable, compared to the mode of operation where the coupling rates are constant, provided the maximum energy that enters the states {right arrow over (V)}<sub>±</sub>, is much less in the variable rate scenario than in the constant rate scenario. In examples, using the proposed scheme of timevarying coupling rates may be advantageous as long as the maximum energy stored in the intermediate resonator is substantially small. In some examples, substantially small may be at most 5% of the peak total energy of the system. In some examples, substantially small may be at most 10% of the peak total energy of the system.
We can now also see why the mode of operation of the system where the coupling rates are kept constant in time may cause a considerable amount of lost (and especially radiated) energy, compared to the proposed mode of operation where the coupling rates are varied adiabatically in time. The reason is that, when κ<sub>1B</sub>=κ<sub>B2</sub>=const, the energies in R<sub>1 </sub>and R<sub>2 </sub>will always be equal to each other if the threeobject system is to stay in {right arrow over (V)}<sub>ds</sub>. So one cannot transfer energy from R<sub>1 </sub>to R<sub>2 </sub>by keeping the system purely in state {right arrow over (V)}<sub>ds</sub>; note that even the initial state of the system, in which all the energy is in R<sub>1 </sub>and there is no energy in R<sub>3</sub>, cannot be solely in {right arrow over (V)}<sub>ds </sub>for fixed nonzero κ<sub>1B </sub>and κ<sub>B2</sub>, and has nonzero components along the eigenstates {right arrow over (V)}<sub>±</sub> which implies a finite energy will build up in R<sub>B</sub>, and consequently result in an increased radiation, especially if Γ<sub>B,rad</sub>>>Γ<sub>A,rad</sub>, which may be the case if the resonator R<sub>B </sub>is chosen large enough to couple strongly to R<sub>1 </sub>and R<sub>2</sub>, as explained earlier.
Illustrative ExampleThe previous analysis explains why a considerable amount of energy was radiated when the inductive coupling rates of the loops were kept constant in time, like in FIG. 17b. Let us now consider the modifications necessary for an adiabaticallyvariedx threeloops indirect transfer scheme, as suggested in the previous section, aiming to reduce the total radiated energy back to its reasonable value in the twoloops direct transfer case (FIG. 17a), while maintaining the total energy transfer at level comparable to the constant(threeloops indirect transfer case (FIG. 17b). In one example, shown in FIG. 17c (Left and Right), we will keep the orientation of L<sub>B </sub>fixed, and start initially (t=0) with L<sub>1 </sub>perpendicular to L<sub>B </sub>for κ<sub>1B</sub>=0 and L<sub>2 </sub>parallel to L<sub>B </sub>for κ<sub>B2</sub>=max, then uniformly rotate L<sub>1 </sub>and L<sub>2</sub>, at the same rates, until finally, at (t=t<sub>EIT</sub>), L<sub>1 </sub>becomes parallel to L<sub>B </sub>for κ<sub>1B</sub>=max and L<sub>2 </sub>perpendicular to L<sub>B </sub>for κ<sub>B2</sub>=0, where we stop the rotation process. In this example, we choose a sinusoidal temporal variation of the coupling rates:
<FORM>κ<sub>1B</sub>(t)=κ sin(πt/2t<sub>EIT</sub>) (44)</FORM>
<FORM>κ<sub>B2</sub>(t)=κ cos (πt/2t<sub>EIT</sub>) (45)</FORM>
for 0<t<t<sub>EIT</sub>, and k<sub>1B</sub>≡κ<sub>1B</sub>/πf=k<sub>B2</sub>=0.0056 as before. By using the same CMT analysis as in Eq. (34), we find, in FIG. 17c (Center), that for an optimal t<sub>EIT*</sub>=1989(1/f), an optimum transfer of {tilde over (η)}<sub>EIT,E*</sub>=61.2% can be achieved at t<sub>*</sub>=1796(1/f), with only 8.2% of the initial energy being radiated, 28.6% absorbed, and 2% left in L<sub>1</sub>. This is quite remarkable: by simply rotating the loops during the transfer, the energy radiated has dropped by factor of 4, while keeping the same 61% level of the energy transferred.
This considerable decrease in radiation may seem surprising, because the intermediate resonator L<sub>B</sub>, which mediates all the energy transfer, is highly radiative (≈650 times more radiative than L<sub>1 </sub>and L<sub>2</sub>), and there is much more time to radiate, since the whole process lasts 14 times longer than in FIG. 17b. Again, the clue to the physical mechanism behind this surprising result can be obtained by observing the differences between the curves describing the energy in R<sub>B </sub>in FIG. 17b and FIG. 17c. Unlike the case of constant coupling rates, depicted in FIG. 17b, where the amount of energy ultimately transferred to L<sub>2 </sub>goes first through the intermediate loop L<sub>B</sub>, with peak energy storage in L<sub>B </sub>as much as 30% of the peak total energy of the system, in the case of timevarying coupling rates, shown in FIG. 17c, there is almost little or no energy in L<sub>B </sub>at all times during the transfer. In other words, the energy is transferred quite efficiently from L<sub>1 </sub>to L<sub>2</sub>, mediated by L<sub>B </sub>without considerable amount of energy ever being in the highly radiative intermediate loop L<sub>B</sub>. (Note that direct transfer from L<sub>1 </sub>to L<sub>2 </sub>is identically zero here since L<sub>1 </sub>is always perpendicular to L<sub>2</sub>, so all the energy transfer is indeed mediated through L<sub>B</sub>). In some examples, improvement in efficiency and/or radiated energy can still have been accomplished if the energy transfer had been designed with a time t<sub>EIT </sub>smaller than its optimal value (perhaps to speed up the process), if the maximum energy accumulated inside the intermediate resonator was less than 30%. For example, improvement can have been achieved for maximum energy accumulation inside the intermediate resonator of 5% or even 10%.
An example implementation of the coupling rate adjustment system 300 is described below, where the resonators are capacitivelyloaded loops, which couple to each other inductively. At the beginning (t=0), the coupling rate adjustment system 300 sets the relative orientation of the first resonator structure L<sub>1 </sub>to be perpendicular to the intermediate resonator structure L<sub>B</sub>. At this orientation, the value of the coupling rate κ<sub>1B </sub>between the first and intermediate resonator structure is at its minimum value. Also, the coupling rate adjustment system 300 can set the relative orientation of the second resonator structure L<sub>2 </sub>to be parallel to the intermediate resonator structure L<sub>B</sub>. At this orientation, the value of the coupling rate κ<sub>B2 </sub>is at a maximum value. During wireless energy transfer, the coupling rate adjustment system 300 can effect the rotation of the first resonator structure L<sub>1 </sub>about its axis so that the value of κ<sub>m </sub>is increased. In some examples, the coupling rate adjustment system 300 can also effect the rotation of the second resonator structure, L<sub>2</sub>, about its axis so that the value of κ<sub>B2 </sub>is decreased. In some examples, a similar effect can be achieved by fixing L<sub>1 </sub>and L<sub>2 </sub>to be perpendicular to each other and rotating only L<sub>B </sub>to be parallel to L<sub>2 </sub>and perpendicular to L<sub>1 </sub>at t=0 and parallel to L<sub>1 </sub>and perpendicular to L<sub>2 </sub>at t=t<sub>EIT</sub>. In some examples, a similar effect can be achieved by fixing L<sub>B </sub>and one of L<sub>1 </sub>and L<sub>2 </sub>(e.g., L<sub>1</sub>) at a predetermined orientation (e.g. at 45 degrees with respect to the intermediate resonator L<sub>B</sub>) and rotating only the other of L<sub>1 </sub>and L<sub>2 </sub>(e.g., L<sub>2 </sub>from parallel to L<sub>B </sub>at t=0 to perpendicular to L<sub>B </sub>at t=t<sub>EIT</sub>).
Similarly, in some implementations, at the beginning (t=0), the coupling rate adjustment system 300 can set the position of the first resonator structure L<sub>1 </sub>at a first large predetermined distance from the intermediate resonator structure L<sub>B </sub>so that the value of the coupling rate κ<sub>1B </sub>is at its minimum value. Correspondingly, the coupling rate adjustment system 300 can set the position of the second resonator structure L<sub>2 </sub>at a second small predetermined distance from the intermediate resonator structure L<sub>B </sub>so that the value of the coupling rate κ<sub>B2 </sub>between the first and intermediate resonator structure is at its maximum value. During wireless energy transfer, the coupling rate adjustment system 300 can affect the position of the first resonator structure L<sub>1 </sub>to bring it closer to L<sub>B </sub>so that the value of κ<sub>1B </sub>is increased. In some examples, the coupling rate adjustment system 300 can also effect the position of the second resonator structure, L<sub>2</sub>, to take it away from L<sub>B </sub>so that the value of κ<sub>B2 </sub>is decreased. In some examples, a similar effect can be achieved by fixing L<sub>1 </sub>and L<sub>2 </sub>to be at a fixed distance to each other and effecting the position of only L<sub>B </sub>to be close to L<sub>2 </sub>and away from L<sub>1 </sub>at t=0 and close to L<sub>1 </sub>and away from L<sub>2 </sub>at t=t<sub>EIT</sub>. In some examples, a similar effect can be achieved by fixing L<sub>B </sub>and one of L<sub>1 </sub>and L<sub>2 </sub>(e.g., L<sub>1</sub>) at a predetermined (not too large but not too small) distance and effecting the position only the other of L<sub>1 </sub>and L<sub>2 </sub>(e.g., L<sub>2 </sub>from close to L<sub>B </sub>at t=0 to away from L<sub>B </sub>at t=t<sub>EIT</sub>).
In the abstract case of energy transfer from R<sub>1 </sub>to R<sub>2</sub>, where no constraints are imposed on the relative magnitude of K, Γ<sub>rad</sub><sup>A</sup>, Γ<sub>rad</sub><sup>B</sup>, Γ<sub>abs</sub><sup>A</sup>, and Γ<sub>abs</sub><sup>B</sup>, it is not certain that the adiabaticκ (EITlike) system will always perform better than the constantκ one, in terms of the transferred and radiated energies. In fact, there could exist some range of the parameters (K, Γ<sub>rad</sub><sup>A</sup>, Γ<sub>rad</sub><sup>B</sup>, Γ<sub>abs</sub><sup>A</sup>, Γ<sub>abs</sub><sup>B</sup>), for which the energy radiated in the constantκtransfer case is less than that radiated in the EITlike case. For this reason, we investigate both the adiabaticκ and constantκ transfer schemes, as we vary some of the crucial parameters of the system. The percentage of energies transferred and lost (radiated+absorbed) depends only on the relative values of κ, Γ<sub>A</sub>=Γ<sub>rad</sub><sup>A </sup>Γ<sub>abs</sub><sup>A </sup>and Γ<sub>B</sub>=Γ<sub>rad</sub><sup>B</sup>+Γ<sub>abs</sub><sup>B</sup>. Hence we first calculate and visualize the dependence of these energies on the relevant parameters κ/Γ<sub>B </sub>and Γ<sub>B</sub>/Γ<sub>A</sub>, in the contour plots shown in FIG. 21.
The way the contour plots are calculated is as follows. For each value of (κ/Γ<sub>B</sub>, Γ<sub>B</sub>/Γ<sub>A</sub>) in the adiabatic case, where κ<sub>1B</sub>(t) and κ<sub>B2</sub>(t) are given by Eq. (44)(45), one tries range of values of t<sub>EIT</sub>. For each t<sub>EIT</sub>, the maximum energy transferred E<sub>2</sub>(%) over 0<t<t<sub>EIT</sub>, denoted by max(E<sub>2</sub>, t<sub>EIT</sub>), is calculated together with the total energy lost at that maximum transfer. Next the maximum of max(E<sub>2</sub>, t<sub>EIT</sub>) over all values of t<sub>EIT </sub>is selected and plotted as single point on the contour plot in FIG. 21a. We refer to this point as the optimum energy transfer (%) in the adiabaticκ case for the particular (κ/Γ<sub>B</sub>, Γ<sub>B</sub>/Γ<sub>A</sub>) under consideration. We also plot in FIG. 21d the corresponding value of the total energy lost (%) at the optimum of E<sub>2</sub>. We repeat these calculations for all pairs (κ/Γ<sub>B</sub>, Γ<sub>B</sub>/Γ<sub>A</sub>) shown in the contour plots. In the constantκ transfer case, for each (κ/Γ<sub>B</sub>, Γ<sub>B</sub>/Γ<sub>A</sub>), the time evolution of E<sub>2</sub>(%) and E<sub>lost </sub>are calculated for 0<t<2/κ, and optimum transfer, shown in FIG. 21b, refers to the maximum of E<sub>3</sub>(t) over 0<t<2/κ. The corresponding total energy lost at optimum constanttransfer is shown in FIG. 21e. Now that we calculated the energies of interest as functions of (κ/Γ<sub>B</sub>, Γ<sub>B</sub>/Γ<sub>A</sub>), we look for ranges of the relevant parameters in which the adiabaticκ transfer has advantages over the constantκ one. So, we plot the ratio of (E<sub>2</sub>)<sub>adiabaticκ</sub>(E<sub>2</sub>)<sub>constantκ</sub> in FIG. 21c, and (E<sub>1ost</sub>)<sub>constantκ</sub>/(E<sub>lost</sub>)<sub>adiabaticκ</sub> in FIG. 21f. We find that, for Γ<sub>B</sub>/Γ<sub>A</sub>>50, the optimum energy transferred in the adiabaticκ case exceeds that in the constantκ case, and the improvement factor can be larger than 2. From FIG. 21f, one sees that the adiabaticκ scheme can reduce the total energy lost by factor of 3 compared to the constantκ scheme, also in the range Γ<sub>B</sub>/Γ<sub>A</sub>>50. As in the constantκ case, also in the adiabaticκ case the efficiency increases as the ratio of the maximum value, K, of the coupling rates to the loss rate of the intermediate object (and thus also the first and second objects for Γ<sub>B</sub>/Γ<sub>A</sub>>1) increases. In some examples, one may design a system so that K is larger than each of Γ<sub>B </sub>and Γ<sub>A</sub>. In some examples, one may design a system so that κ is at least 2 times larger than each of Γ<sub>B </sub>and Γ<sub>A</sub>. In some examples, one may design a system so that κ is at least 4 times larger than each of Γ<sub>B </sub>and Γ<sub>A</sub>.
Although one may be interested in reducing the total energy lost (radiated+absorbed) as much as possible in order to make the transfer more efficient, the undersirable nature of the radiated energy may make it important to consider reducing the energy radiated. For this purpose, we calculate the energy radiated at optimum transfer in both the adiabaticκ and constantκ schemes, and compare them. The relevant parameters in this case are κ/Γ<sub>B</sub>, Γ<sub>B</sub>/Γ<sub>A</sub>, Γ<sub>rad</sub><sup>A</sup>/Γ<sub>A</sub>, and Γ<sub>rad</sub><sup>B</sup>/Γ<sub>B</sub>. The problem is more complex because the parameter space is now 4dimensional. So we focus on those particular cross sections that can best reveal the most important differences between the two schemes. From FIGS. 21c and 21f, one can guess that the best improvement in both E<sub>2 </sub>and E<sub>lost </sub>occurs for Γ<sub>B</sub>/Γ<sub>A</sub>≧500. Moreover, knowing that it is the intermediate object R<sub>B </sub>that makes the main difference between the adiabaticκ and constantκ schemes, being “energyempty” in the adiabaticκ case and “energyfull” in the constantκ one, we first look at the special situation where Γ<sub>rad</sub><sup>A</sup>=0. In FIG. 22a and FIG. 22b, we show contour plots of the energy radiated at optimum transfer, in the constantκ and adiabaticκschemes respectively, for the particular cross section having Γ<sub>B</sub>/Γ<sub>A</sub>=500 and Γ<sub>rad</sub><sup>A</sup>=0. Comparing these two figures, one can see that, by using the adiabaticκ scheme, one can reduce the energy radiated by factor of 6.3 or more.
To get quantitative estimate of the radiation reduction factor in the general case where Γ<sub>A,rad</sub>≠0, we calculate the ratio of energies radiated at optimum transfers in both schemes, namely,
<maths id="MATHUS00044" num="00044"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><msub><mrow><mo>(</mo><msub><mi>E</mi><mi>rad</mi></msub><mo>)</mo></mrow><mrow><mi>constant</mi><mo></mo><mi>κ</mi></mrow></msub><msub><mrow><mo>(</mo><msub><mi>E</mi><mi>rad</mi></msub><mo>)</mo></mrow><mrow><mi>adiabatic</mi><mo></mo><mi>κ</mi></mrow></msub></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mo></mo><mrow><msubsup><mo>∫</mo><mn>0</mn><msubsup><mi>t</mi><mo>*</mo><mrow><mi>constant</mi><mo></mo><mi>κ</mi></mrow></msubsup></msubsup><mo></mo><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mrow><mfrac><msubsup><mi>Γ</mi><mi>rad</mi><mi>B</mi></msubsup><msubsup><mi>Γ</mi><mi>rad</mi><mi>A</mi></msubsup></mfrac><mo></mo><msup><mrow><mo></mo><mrow><msubsup><mi>a</mi><mi>B</mi><mrow><mi>constant</mi><mo></mo><mi>κ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo></mrow><mn>2</mn></msup></mrow><mo>+</mo></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mo></mo><mrow><msubsup><mi>a</mi><mn>1</mn><mrow><mi>constant</mi><mo></mo><mi>κ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo></mrow><mn>2</mn></msup><mo>+</mo></mrow></mtd></mtr><mtr><mtd><msup><mrow><mo></mo><mrow><msubsup><mi>a</mi><mn>2</mn><mrow><mi>constant</mi><mo></mo><mi>κ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo></mrow><mn>2</mn></msup></mtd></mtr></mtable><mo>}</mo></mrow></mrow></mrow><mrow><mn>2</mn><mo></mo><mrow><msubsup><mo>∫</mo><mn>0</mn><msubsup><mi>t</mi><mo>*</mo><mrow><mi>adiabatic</mi><mo></mo><mi>κ</mi></mrow></msubsup></msubsup><mo></mo><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mrow><mfrac><msubsup><mi>Γ</mi><mi>rad</mi><mi>B</mi></msubsup><msubsup><mi>Γ</mi><mi>rad</mi><mi>A</mi></msubsup></mfrac><mo></mo><msup><mrow><mo></mo><mrow><msubsup><mi>a</mi><mi>B</mi><mrow><mi>adiabatic</mi><mo></mo><mi>κ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo></mrow><mn>2</mn></msup></mrow><mo>+</mo></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mo></mo><mrow><msubsup><mi>a</mi><mn>1</mn><mrow><mi>adiabatic</mi><mo></mo><mi>κ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo></mrow><mn>2</mn></msup><mo>+</mo></mrow></mtd></mtr><mtr><mtd><msup><mrow><mo></mo><mrow><msubsup><mi>a</mi><mn>2</mn><mrow><mi>adiabatic</mi><mo></mo><mi>κ</mi></mrow></msubsup><mo></mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo></mo></mrow><mn>2</mn></msup></mtd></mtr></mtable><mo>}</mo></mrow></mrow></mrow></mfrac></mrow></mtd><mtd><mrow><mo>(</mo><mn>46</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
which depends only on Γ<sub>rad</sub><sup>B</sup>/Γ<sub>rad</sub><sup>A</sup>, the timedependent mode amplitudes, and the optimum transfer times in both schemes. The latter two quantities are completely determined by κ/Γ<sub>B </sub>and Γ<sub>B</sub>/Γ<sub>A</sub>. Hence the only parameters relevant to the calculations of the ratio of radiated energies are Γ<sub>rad</sub><sup>B</sup>/Γ<sub>rad</sub><sup>A</sup>, κ/Γ<sub>B </sub>and Γ<sub>B</sub>/Γ<sub>A</sub>, thus reducing the dimensionality of the investigated parameter space from down to 3. For convenience, we multiply the first relevant parameter Γ<sub>rad</sub><sup>B</sup>/Γ<sub>rad</sub><sup>A </sup>by Γ<sub>B</sub>/Γ<sub>A </sub>which becomes (Γ<sub>rrad</sub><sup>B</sup>/Γ<sub>B</sub>)/(Γ<sub>rad</sub><sup>A</sup>/Γ<sub>A</sub>), i.e. the ratio of quantities that specify what percentage of each object's loss is radiated. Next, we calculate the ratio of energies radiated as function of (Γ<sub>rad</sub><sup>B</sup>/Γ<sub>B</sub>)/(Γ<sub>rad</sub><sup>A</sup>/Γ<sub>A</sub>) and κ/Γ<sub>B </sub>in the two special cases Γ<sub>B</sub>/Γ<sub>A</sub>=50, and Γ<sub>B</sub>/Γ<sub>A</sub>=500, and plot them in FIG. 22c and FIG. 22d, respectively. We also show, in FIG. 22e, the dependence of (E<sub>rad</sub>)<sub>constantκ</sub>/(E<sub>rad</sub>)<sub>EITlike </sub>on κ/Γ<sub>B </sub>and Γ<sub>B</sub>/Γ<sub>A</sub>, for the special case Γ<sub>rad</sub><sup>A</sup>=0. As can be seen from FIGS. 22c22d, the adiabaticκ scheme is less radiative than the constantκ scheme whenever Γ<sub>rad</sub><sup>B</sup>/Γ<sub>B </sub>is larger than Γ<sub>rad</sub><sup>A</sup>/Γ<sub>A</sub>, and the radiation reduction ratio increases as Γ<sub>B</sub>/Γ<sub>A </sub>and κ/Γ<sub>B </sub>are increased (see FIG. 22e). In some examples, the adiabaticκ scheme is less radiative than the constantκ scheme whenever Γ<sub>rad</sub><sup>B</sup>/Γ<sub>rad</sub><sup>A </sup>is larger than about 20. In some examples, the adiabaticκ scheme is less radiative than the constantκ scheme whenever Γ<sub>rad</sub><sup>B</sup>/Γ<sub>rad</sub><sup>A </sup>is larger than about 50.
It is to be understood that while three resonant objects are shown in the previous examples, other examples can feature four or more resonant objects. For example, in some examples, a single source object can transfer energy to multiple device objects through one intermediate object. In some examples, energy can be transferred from a source resonant object to a device resonant object, through two or more intermediate resonant objects, and so forth.
In general, the overall performance of an example of the resonancebased wireless energytransfer scheme depends strongly on the robustness of the resonant objects' resonances. Therefore, it is desirable to analyze the resonant objects' sensitivity to the near presence of random nonresonant extraneous objects. One appropriate analytical model is that of “perturbation theory” (PT), which suggests that in the presence of an extraneous perturbing object p the field amplitude a<sub>1</sub>(t) inside the resonant object 1 satisfies, to first order:
<maths id="MATHUS00045" num="00045"><math overflow="scroll"><mtable><mtr><mtd><mrow><mfrac><mrow><mo></mo><msub><mi>a</mi><mn>1</mn></msub></mrow><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo>=</mo><mrow><mrow><mrow><mo></mo><mrow><mi></mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>ω</mi><mn>1</mn></msub><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mn>1</mn></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><msub><mi>a</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><mrow><mi></mi><mo></mo><mrow><mo>(</mo><mrow><mrow><mi>δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mrow><mn>11</mn><mo></mo><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></msub></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mrow><mn>1</mn><mo></mo><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow></msub></mrow></mrow><mo>)</mo></mrow></mrow><mo></mo><msub><mi>a</mi><mn>1</mn></msub></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>47</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where again ω<sub>1 </sub>is the frequency and Γ<sub>1 </sub>the intrinsic (absorption, radiation etc.) loss rate, while δκ<sub>11(p) </sub>is the frequency shift induced onto 1 due to the presence of p and δΓ<sub>1(p) </sub>is the extrinsic due to p (absorption inside p, scattering from p etc.) loss rate. δΓ<sub>1(p) </sub>is defined as ΓΓ<sub>1(p)</sub>≡Γ<sub>1(p)</sub>−Γ<sub>1</sub>, where Γ<sub>1(p) </sub>is the total perturbed loss rate in the presence of p. The firstorder PT model is valid only for small perturbations. Nevertheless, the parameters δκ<sub>11(p)</sub>, δΓ<sub>1(p) </sub>are well defined, even outside that regime, if a<sub>1 </sub>is taken to be the amplitude of the exact perturbed mode. Note also that interference effects between the radiation field of the initial resonantobject mode and the field scattered off the extraneous object can for strong scattering (e.g. off metallic objects) result in total Γ<sub>1,rad(p) </sub>that are smaller than the initial Γ<sub>1,rad </sub>(namely δΓ<sub>1,rad(p) </sub>is negative).
It has been shown that a specific relation is desired between the resonant frequencies of the source and deviceobjects and the driving frequency. In some examples, all resonant objects must have the same eigenfrequency and this must be equal to the driving frequency. In some implementations, this frequencyshift can be “fixed” by applying to one or more resonant objects and the driving generator a feedback mechanism that corrects their frequencies. In some examples, the driving frequency from the generator can be fixed and only the resonant frequencies of the objects can be tuned with respect to this driving frequency.
The resonant frequency of an object can be tuned by, for example, adjusting the geometric properties of the object (e.g. the height of a selfresonant coil, the capacitor plate spacing of a capacitivelyloaded loop or coil, the dimensions of the inductor of an inductivelyloaded rod, the shape of a dielectric disc, etc.) or changing the position of a nonresonant object in the vicinity of the resonant object.
In some examples, referring to FIG. 23a, each resonant object is provided with an oscillator at fixed frequency and a monitor which determines the eigenfrequency of the object. At least one of the oscillator and the monitor is coupled to a frequency adjuster which can adjust the frequency of the resonant object. The frequency adjuster determines the difference between the fixed driving frequency and the object frequency and acts, as described above, to bring the object frequency into the required relation with respect to the fixed frequency. This technique assures that all resonant objects operate at the same fixed frequency, even in the presence of extraneous objects.
In some examples, referring to FIG. 23(b), during energy transfer from a source object to a device object, the device object provides energy or power to a load, and an efficiency monitor measures the efficiency of the energytransfer or powertransmission. A frequency adjuster coupled to the load and the efficiency monitor acts, as described above, to adjust the frequency of the object to maximize the efficiency.
In other examples, the frequency adjusting scheme can rely on information exchange between the resonant objects. For example, the frequency of a source object can be monitored and transmitted to a device object, which is in turn synched to this frequency using frequency adjusters, as described above. In other embodiments the frequency of a single clock can be transmitted to multiple devices, and each device then synched to that frequency using frequency adjusters, as described above.
Unlike the frequency shift, the extrinsic perturbing loss due to the presence of extraneous perturbing objects can be detrimental to the functionality of the energytransfer scheme, because it is difficult to remedy. Therefore, the total perturbed quality factors Q<sub>(p) </sub>(and the corresponding perturbed strongcoupling factor U<sub>(p) </sub>should be quantified.
In some examples, a system for wireless energytransfer uses primarily magnetic resonances, wherein the energy stored in the near field in the air region surrounding the resonator is predominantly magnetic, while the electric energy is stored primarily inside the resonator. Such resonances can exist in the quasistatic regime of operation (r<d) that we are considering: for example, for coils with h<<2r, most of the electric field is localized within the selfcapacitance of the coil or the externally loading capacitor and, for dielectric disks, with ∈>>1 the electric field is preferentially localized inside the disk. In some examples, the influence of extraneous objects on magnetic resonances is nearly absent. The reason is that extraneous nonconducting objects p that could interact with the magnetic field in the air region surrounding the resonator and act as a perturbation to the resonance are those having significant magnetic properties (magnetic permeability Re{u}>1 or magnetic loss Im{μ}>0). Since almost all everyday nonconducting materials are nonmagnetic but just dielectric, they respond to magnetic fields in the same way as free space, and thus will not disturb the resonance of the resonator. Extraneous conducting materials can however lead to some extrinsic losses due to the eddy currents induced inside them or on their surface (depending on their conductivity). However, even for such conducting materials, their presence will not be detrimental to the resonances, as long as they are not in very close proximity to the resonant objects.
The interaction between extraneous objects and resonant objects is reciprocal, namely, if an extraneous object does not influence a resonant object, then also the resonant object does not influence the extraneous object. This fact can be viewed in light of safety considerations for human beings. Humans are also nonmagnetic and can sustain strong magnetic fields without undergoing any risk. A typical example, where magnetic fields B˜1T are safely used on humans, is the Magnetic Resonance Imaging (MRI) technique for medical testing. In contrast, the magnetic nearfield required in typical embodiments in order to provide a few Watts of power to devices is only B˜10<sup>−4</sup>T, which is actually comparable to the magnitude of the Earth's magnetic field. Since, as explained above, a strong electric nearfield is also not present and the radiation produced from this nonradiative scheme is minimal, the energytransfer apparatus, methods and systems described herein is believed safe for living organisms.
An advantage of the presently proposed technique using adiabatic variations of the coupling rates between the first and intermediate resonators and between the intermediate and second resonators compared to a mode of operation where these coupling rates are not varied but are constant is that the interactions of the intermediate resonator with extraneous objects can be greatly reduced with the presently proposed scheme. The reason is once more the fact that there is always a substantially small amount of energy in the intermediate resonator in the adiabaticκ scheme, therefore there is little energy that can be induced from the intermediate object to an extraneous object in its vicinity. Furthermore, since the losses of the intermediate resonator are substantially avoided in the adiabaticκ case, the system is less immune to potential reductions of the quality factor of the intermediate resonator due to extraneous objects in its vicinity.
In some examples, one can estimate the degree to which the resonant system of a capacitivelyloaded conductingwire coil has mostly magnetic energy stored in the space surrounding it. If one ignores the fringing electric field from the capacitor, the electric and magnetic energy densities in the space surrounding the coil come just from the electric and magnetic field produced by the current in the wire; note that in the far field, these two energy densities must be equal, as is always the case for radiative fields. By using the results for the fields produced by a subwavelength (r<<λ) current loop (magnetic dipole) with h=0, we can calculate the ratio of electric to magnetic energy densities, as a function of distance D<sub>p </sub>from the center of the loop (in the limit r<<D<sub>p</sub>) and the angle θ with respect to the loop axis:
<maths id="MATHUS00046" num="00046"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mrow><mtable><mtr><mtd><mrow><mfrac><mrow><msub><mi>w</mi><mi>e</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><msub><mi>w</mi><mi>m</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mi/><mo></mo><mfrac><mrow><msub><mi>ɛ</mi><mi>o</mi></msub><mo></mo><msup><mrow><mo></mo><mrow><mi>E</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo></mo></mrow><mn>2</mn></msup></mrow><mrow><msub><mi>μ</mi><mi>o</mi></msub><mo></mo><msup><mrow><mo></mo><mrow><mi>H</mi><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo></mo></mrow><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>=</mo><mi/><mo></mo><mfrac><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></mrow><mo>)</mo></mrow><mo></mo><msup><mi>sin</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow><mtable><mtr><mtd><mrow><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>4</mn></msup></mfrac></mrow><mo>)</mo></mrow><mo></mo><mn>4</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>cos</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow><mo>+</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo></mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>4</mn></msup></mfrac></mrow><mo>)</mo></mrow><mo></mo><msup><mi>sin</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow></mtd></mtr></mtable></mfrac></mrow><mo>;</mo><mrow><mi>x</mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>π</mi><mo></mo><mfrac><msub><mi>D</mi><mi>p</mi></msub><mi>λ</mi></mfrac></mrow></mrow></mrow></mtd></mtr></mtable><mo></mo><mstyle><mtext></mtext></mstyle><mo>⇒</mo><mfrac><mrow><munder><mo>∯</mo><msub><mi>S</mi><mi>p</mi></msub></munder><mo></mo><mrow><mrow><msub><mi>w</mi><mi>e</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo></mo><mi>S</mi></mrow></mrow></mrow><mrow><munder><mo>∯</mo><msub><mi>S</mi><mi>p</mi></msub></munder><mo></mo><mrow><mrow><msub><mi>w</mi><mi>m</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo></mo><mrow><mo></mo><mi>S</mi></mrow></mrow></mrow></mfrac></mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></mrow><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mn>3</mn><msup><mi>x</mi><mn>4</mn></msup></mfrac></mrow></mfrac></mrow><mo>;</mo><mrow><mi>x</mi><mo>=</mo><mrow><mn>2</mn><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>π</mi><mo></mo><mfrac><msub><mi>D</mi><mi>p</mi></msub><mi>λ</mi></mfrac></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>48</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where the second line is the ratio of averages over all angles by integrating the electric and magnetic energy densities over the surface of a sphere of radius D<sub>p</sub>. From Eq. (48) it is obvious that indeed for all angles in the near field (x<<1) the magnetic energy density is dominant, while in the far field (x>>1) they are equal as they should be. Also, the preferred positioning of the loop is such that objects which can interfere with its resonance lie close to its axis (θ=0), where there is no electric field. For example, using the systems described in Table 4, we can estimate from Eq. (48) that for the loop of r=30 cm at a distance D<sub>P</sub>=10r=3m the ratio of average electric to average magnetic energy density would be ˜12% and at D<sub>p</sub>=3r=90 cm it would be ˜1%, and for the loop of r=10 cm at a distance D<sub>p</sub>=10r=1m the ratio would be ˜33% and at D<sub>p</sub>=3r=30 cm it would be ˜2.5%. At closer distances this ratio is even smaller and thus the energy is predominantly magnetic in the near field, while in the radiative far field, where they are necessarily of the same order (ratio→1) both are very small, because the fields have significantly decayed, as capacitivelyloaded coil systems are designed to radiate very little. Therefore, this is the criterion that qualifies this class of resonant system as a magnetic resonant system.
To provide an estimate of the effect of extraneous objects on the resonance of a capacitivelyloaded loop including the capacitor fringing electric field, we use the perturbation theory formula, stated earlier, δΓ<sub>1,abs(p)</sub>=ω<sub>1</sub>/4·∫d<sup>3</sup>rIm{∈<sub>p</sub>(r)}E<sub>1</sub>(r)(r)<sup>2</sup>/W with the computational FEFD results for the field of an example like the one shown in the plot of FIG. 5 and with a rectangular object of dimensions 30 cm×30 cm×1.5m and permittivity ∈=49+16i (consistent with human muscles) residing between the loops and almost standing on top of one capacitor (˜3 cm away from it) and find δQ<sub>abs(human)</sub>˜10<sup>5 </sup>and for ˜10 cm away δQ<sub>abs(human)</sub>˜5˜10<sup>5</sup>. Thus, for ordinary distances (˜1m) and placements (not immediately on top of the capacitor) or for most ordinary extraneous objects p of much smaller losstangent, we conclude that it is indeed fair to say that δQ<sub>abs(p)</sub>∞. The only perturbation that is expected to affect these resonances is a close proximity of large metallic structures.
Selfresonant coils can be more sensitive than capacitivelyloaded coils, since for the former the electric field extends over a much larger region in space (the entire coil) rather than for the latter (just inside the capacitor). On the other hand, selfresonant coils can be simple to make and can withstand much larger voltages than most lumped capacitors. Inductivelyloaded conducting rods can also be more sensitive than capacitivelyloaded coils, since they rely on the electric field to achieve the coupling.
For dielectric disks, small, lowindex, lowmaterialloss or faraway stray objects will induce small scattering and absorption. In such cases of small perturbations these extrinsic loss mechanisms can be quantified using respectively the analytical firstorder perturbation theory formulas
<FORM>[δQ<sub>1,rad(p)</sub>]<sup>−1</sup>≡2δΓ<sub>1,rad(p)</sub>/ω<sub>1</sub>∝∫d<sup>3</sup>r[Re{∈<sub>p</sub>(r)}E<sub>1</sub>(r)]<sup>2</sup>/W </FORM>
<FORM>[δQ<sub>1,abs(p)</sub>]<sup>−1≡2δΓ</sup><sub>1,abs(p)</sub>/ω<sub>1</sub>∝∫d<sup>3</sup>rIm{∈<sub>p</sub>(r)}E<sub>1</sub>(r)<sup>2</sup>/2W </FORM>
where W=∫d<sup>3</sup>r∈(r)E<sub>1</sub>(r)<sup>2</sup>/2 is the total resonant electromagnetic energy of the unperturbed mode. As one can see, both of these losses depend on the square of the resonant electric field tails E<sub>1 </sub>at the site of the extraneous object. In contrast, the coupling factor from object 1 to another resonant object 2 is, as stated earlier,
<FORM>k<sub>12</sub>=2κ<sub>12</sub>/√{square root over (ω<sub>1</sub>ω<sub>2</sub>)}≈∫d<sup>3</sup>r∈<sub>2</sub>(r)E<sub>2</sub>*(r)E<sub>1</sub>(r)/∫d<sup>3</sup>r∈(r)E<sub>1</sub>(r)<sup>2 </sup></FORM>
and depends linearly on the field tails E<sub>1 </sub>of 1 inside 2. This difference in scaling gives us confidence that, for, for example, exponentially small field tails, coupling to other resonant objects should be much faster than all extrinsic loss rates(κ<sub>12</sub>>>δΓ<sub>1,2(p)</sub>), at least for small perturbations, and thus the energytransfer scheme is expected to be sturdy for this class of resonant dielectric disks.
However, we also want to examine certain possible situations where extraneous objects cause perturbations too strong to analyze using the above firstorder perturbation theory approach. For example, we place a dielectric disk close to another offresonance object of large Re{∈}, Im{∈} and of same size but different shape (such as a human being h), as shown in FIG. 24a, and a roughened surface of large extent but of small Re{∈}, Im{∈} (such as a wall w), as shown in FIG. 24b. For distances D<sub>h,w</sub>/r=103 between the diskcenter and the “human”center or “wall”, the numerical FDFD simulation results presented in FIGS. 24a and 24b suggest that, the disk resonance seems to be fairly robust, since it is not detrimentally disturbed by the presence of extraneous objects, with the exception of the very close proximity of highloss objects. To examine the influence of large perturbations on an entire energytransfer system we consider two resonant disks in the close presence of both a “human” and a “wall”. Comparing Table 8 to the table in FIG. 24c, the numerical FDFD simulations show that the system performance deteriorates from U˜150 to U<sub>(hw)</sub>˜0.510, i.e. only by acceptably small amounts.
In general, different examples of resonant systems have different degree of sensitivity to external perturbations, and the resonant system of choice depends on the particular application at hand, and how important matters of sensitivity or safety are for that application. For example, for a medical implantable device (such as a wirelessly powered artificial heart) the electric field extent must be minimized to the highest degree possible to protect the tissue surrounding the device. In such cases where sensitivity to external objects or safety is important, one should design the resonant systems so that the ratio of electric to magnetic energy density w<sub>e</sub>/w<sub>m </sub>is reduced or minimized at most of the desired (according to the application) points in the surrounding space.
The nonradiative wireless energy transfer techniques described above can enable efficient wireless energyexchange between resonant objects, while suffering only modest transfer and dissipation of energy into other extraneous offresonant objects. The technique is general, and can be applied to a variety of resonant systems in nature. In this Section, we identify a variety of applications that can benefit from or be designed to utilize wireless power transmission.
Remote devices can be powered directly, using the wirelessly supplied power or energy to operate or run the devices, or the devices can be powered by or through or in addition to a battery or energy storage unit, where the battery is occasionally being charged or recharged wirelessly. The devices can be powered by hybrid battery/energy storage devices such as batteries with integrated storage capacitors and the like. Furthermore, novel battery and energy storage devices can be designed to take advantage of the operational improvements enabled by wireless power transmission systems.
Devices can be turned off and the wirelessly supplied power or energy used to charge or recharge a battery or energy storage unit. The battery or energy storage unit charging or recharging rate can be high or low. The battery or energy storage units can be trickle charged or float charged. It would be understood by one of ordinary skill in the art that there are a variety of ways to power and/or charge devices, and the variety of ways could be applied to the list of applications that follows.
Some wireless energy transfer examples that can have a variety of possible applications include for example, placing a source (e.g. one connected to the wired electricity network) on the ceiling of a room, while devices such as robots, vehicles, computers, PDAs or similar are placed or move freely within the room. Other applications can include powering or recharging electricengine buses and/or hybrid cars and medical implantable devices. Additional example applications include the ability to power or recharge autonomous electronics (e.g. laptops, cellphones, portable music players, household robots, GPS navigation systems, displays, etc), sensors, industrial and manufacturing equipment, medical devices and monitors, home appliances (e.g. lights, fans, heaters, displays, televisions, countertop appliances, etc.), military devices, heated or illuminated clothing, communications and navigation equipment, including equipment built into vehicles, clothing and protectivewear such as helmets, body armor and vests, and the like, and the ability to transmit power to physically isolated devices such as to implanted medical devices, to hidden, buried, implanted or embedded sensors or tags, to and/or from rooftop solar panels to indoor distribution panels, and the like.
A number of examples of the invention have been described. Nevertheless, it will be understood that various modifications can be made without departing from the spirit and scope of the invention.