ADAPTIVE WIRELESS POWER TRANSFER APPARATUS AND METHOD THEREOF

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2
Assignments
First Claim
1. A method comprising:
 coupling a highQ resonator of a transmitter and a highQ resonator of a receiver together by a common inductance of the transmitter and the receiver; and
adjusting the transmitter, the receiver, or both, to control power transmitted wirelessly between the transmitter and the receiver.
2 Assignments
0 Petitions
Accused Products
Abstract
Described herein are embodiments of a method that includes coupling a highQ resonator of a transmitter and a highQ resonator of a receiver together by a common inductance of the transmitter and the receiver; and adjusting the transmitter, the receiver, or both, to control power transmitted wirelessly between the transmitter and the receiver.
263 Citations
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Current Assignee
Massachusetts Institute of Technology

Sponsoring Entity
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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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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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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Witricity Corporation

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Witricity Corporation

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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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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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Current Assignee
Massachusetts Institute of Technology

Sponsoring Entity
Massachusetts Institute of Technology

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Auckland UniServices Limited, Daifuku Company Limited

Sponsoring Entity
Auckland UniServices Limited, Daifuku Company Limited

Oscillatorshuttlecircuit (OSC) networks for conditioning energy in higherorder symmetry algebraic topological forms and RF phase conjugation  
Patent #
US 5,493,691 A
Filed 12/23/1993

Current Assignee
BARRETT HOLDING LLC

Sponsoring Entity
Barrett Terence W.

Inductive power pickup coils  
Patent #
US 5,528,113 A
Filed 10/21/1994

Current Assignee
Auckland UniServices Limited

Sponsoring Entity
Auckland UniServices Limited

Separable inductive coupler  
Patent #
US 5,216,402 A
Filed 01/22/1992

Current Assignee
General Motors Corporation

Sponsoring Entity
Hughes Aircraft Company

Dual feedback control for a highefficiency classd power amplifier circuit  
Patent #
US 5,118,997 A
Filed 08/16/1991

Current Assignee
General Electric Company

Sponsoring Entity
General Electric Company

RF beam center location method and apparatus for power transmission system  
Patent #
US 4,088,999 A
Filed 05/21/1976

Current Assignee
Fletcher James C. Administrator of the National Aeronautics and Space Administration with respect to an invention of, Dickinson Richard M.

Sponsoring Entity
Fletcher James C. Administrator of the National Aeronautics and Space Administration with respect to an invention of, Dickinson Richard M.

ENERGY TRANSLATING DEVICE  
Patent #
US 3,517,350 A
Filed 07/07/1969

Current Assignee
Beaver William D.

Sponsoring Entity
Beaver William D.

17 Claims
 1. A method comprising:
 coupling a highQ resonator of a transmitter and a highQ resonator of a receiver together by a common inductance of the transmitter and the receiver; and
adjusting the transmitter, the receiver, or both, to control power transmitted wirelessly between the transmitter and the receiver.  View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13)
 coupling a highQ resonator of a transmitter and a highQ resonator of a receiver together by a common inductance of the transmitter and the receiver; and
 14. A transmitter comprising:
 a highQ resonator configured to couple with a highQ resonator of a receiver by a common inductance; and
a controller configured to optimize transmitted power, the transmitter configured to transmit power wirelessly to the receiver.  View Dependent Claims (15)
 a highQ resonator configured to couple with a highQ resonator of a receiver by a common inductance; and
 16. A receiver comprising:
 a highQ resonator configured to couple with a highQ resonator of a transmitter by a common inductance; and
optimally receive power wirelessly from the transmitter.  View Dependent Claims (17)
 a highQ resonator configured to couple with a highQ resonator of a transmitter by a common inductance; and
1 Specification
This application is a continuation of U.S. patent application Ser. No. 12/688,339 ('339 application) filed Jan. 15, 2010, the entirety of which is incorporated herein by reference. The '339 application is a continuation to U.S. patent application Ser. No. 12/055,963 ('963 Application), filed Mar. 26, 2008 the entirety of which is incorporated herein by reference. The '963 application claims the benefit of the following provisional applications, each of which is incorporated herein by reference in its entirety: U.S. Provisional Patent Application 60/908,383 filed Mar. 27, 2007; and U.S. Provisional Patent Application 60/908,666, filed Mar. 28, 2007.
The '963 application is a continuationinpart of copending United States patent application entitled WIRELESS NONRADIATIVE ENERGY TRANSFER filed on Jul. 5, 2006 and having Ser. No. 11/481,077 ('077 Application), the entirety of which is incorporated herein by reference. The '077 Application claims the benefit of provisional application Ser. No. 60/698,442 filed Jul. 12, 2005 ('442 Application), the entirety of which is incorporated herein by reference.
The '963 application, pursuant to U.S.C. §120 and U.S.C. §363, is a continuationinpart of International Application No. PCT/US2007/070892, filed Jun. 11, 2007, which is incorporated herein by reference in its entirety, and which claims priority to the following provisional applications, each of which is incorporated herein by reference in its entirety: U.S. Provisional Patent Application 60/908,383 filed Mar. 27, 2007; and U.S. Provisional Patent Application 60/908,666, filed Mar. 28, 2007.
STATEMENT REGARDING GOVERNMENT FUNDINGThis invention was made with government support awarded by the National Science Foundation under Grant No. DMR 0213282. The government has certain rights in this invention.
BACKGROUNDThe disclosure relates to wireless energy transfer. Wireless energy transfer may 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><img id="CUSTOMCHARACTER00001" he="2.79mm" wi="1.44mm" file="US20110140544A120110616P00001.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>L<sub>DEV</sub>, where L<sub>DEV </sub>is the characteristic size of the device and/or the source), but 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><img id="CUSTOMCHARACTER00002" he="2.46mm" wi="1.44mm" file="US20110140544A120110616P00002.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>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.
SUMMARYThe inventors have realized that resonant objects with coupled resonant modes having localized evanescent field patterns may be used for nonradiative wireless energy transfer. Resonant objects tend to couple, while interacting weakly with other offresonant environmental objects. Typically, using the techniques described below, as the coupling increases, so does the transfer efficiency. In some embodiments, using the below techniques, the energytransfer rate can be larger than the energyloss rate. Accordingly, efficient wireless energyexchange can be achieved between the resonant objects, while suffering only modest transfer and dissipation of energy into other offresonant objects. The nearlyomnidirectional but stationary (nonlossy) nature of the near field makes this mechanism suitable for mobile wireless receivers. Various embodiments therefore have a variety of possible applications including for example, placing a source (e.g. one connected to the wired electricity network) on the ceiling of a factory room, while devices (robots, vehicles, computers, or similar) are roaming freely within the room. Other applications include power supplies for electricengine buses and/or hybrid cars and medical implantable devices.
In some embodiments, resonant modes are socalled magnetic resonances, for which most of the energy surrounding the resonant objects is stored in the magnetic field; i.e. there is very little electric field outside of the resonant objects. Since most everyday materials (including animals, plants and humans) are nonmagnetic, their interaction with magnetic fields is minimal. This is important both for safety and also to reduce interaction with the extraneous environmental objects.
In one aspect, an apparatus is disclosed for use in wireless energy transfer, which includes a first resonator structure configured to transfer energy with a second resonator structure over a distance D greater than a characteristic size L<sub>2 </sub>of the second resonator structure. In some embodiments, D is also greater than one or more of: a characteristic size L<sub>1 </sub>of the first resonator structure, a characteristic thickness T<sub>1 </sub>of the first resonator structure, and a characteristic width W<sub>1 </sub>of the first resonator structure. The energy transfer is mediated by evanescenttail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure. The apparatus may include any of the following features alone or in combination.
In some embodiments, the first resonator structure is configured to transfer energy to the second resonator structure. In some embodiments, the first resonator structure is configured to receive energy from the second resonator structure. In some embodiments, the apparatus includes the second resonator structure.
In some embodiments, the first resonator structure has a resonant angular frequency ω<sub>1</sub>, a Qfactor Q<sub>1</sub>, and a resonance width Γ<sub>1</sub>, the second resonator structure has a resonant angular frequency ω<sub>2</sub>, a Qfactor Q<sub>2</sub>, and a resonance width Γ<sub>2 </sub>and the energy transfer has a rate κ. In some embodiments, the absolute value of the difference of the angular frequencies ω<sub>1 </sub>and ω<sub>2 </sub>is smaller than the broader of the resonant widths Γ<sub>1 </sub>and Γ<sub>2</sub>.
In some embodiments Q<sub>1</sub>>100 and Q<sub>2</sub>>100, Q<sub>1</sub>>300 and Q<sub>2</sub>>300, Q<sub>1</sub>>500 and Q<sub>2</sub>>500, or Q<sub>1</sub>>1000 and Q<sub>2</sub>>1000. In some embodiments, Q<sub>1</sub>>100 or Q<sub>2</sub>>100, Q<sub>1</sub>>300 or Q<sub>2</sub>>300, Q<sub>1</sub>>500 or Q<sub>2</sub>>500, or Q<sub>1</sub>>1000 or Q<sub>2</sub>>1000.
In some embodiments, the coupling to loss ratio
<maths id="MATHUS00001" num="00001"><math overflow="scroll"><mrow><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><mn>0.5</mn></mrow><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><mn>1</mn></mrow><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><mn>2</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>></mo><mn>5.</mn></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>2 </sub>may be as large as 2, as large as 3, as large as 5, as large as 7, or as large as 10.
In some embodiments, Q<sub>1</sub>>1000 and Q<sub>2</sub>>1000, and the coupling to loss ratio
<maths id="MATHUS00002" num="00002"><math overflow="scroll"><mrow><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><mn>10</mn></mrow><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><mn>25</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>></mo><mn>40.</mn></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>2 </sub>may be as large as 2, as large as 3, as large as 5, as large as 7, as large as 10.
In some embodiments, Q<sub>κ</sub>=ω/2K is less than about 50, less than about 200, less than about 500, or less than about 1000. In some such embodiments, D/L<sub>2 </sub>is as large as 2, as large as 3, as large as 5, as large as 7, or as large as 10.
In some embodiments, the quantity κ/√{square root over (Γ<sub>1</sub>Γ<sub>1</sub>)} is maximized at an angular frequency {tilde over (ω)} with a frequency width {tilde over (Γ)} around the maximum, and the absolute value of the difference of the angular frequencies ω<sub>1 </sub>and {tilde over (ω)} is smaller than the width {tilde over (Γ)}, and the absolute value of the difference of the angular frequencies ω<sub>2 </sub>and {tilde over (ω)} is smaller than the width {tilde over (Γ)}.
In some embodiments, the energy transfer operates with an efficiency η<sub>work </sub>greater than about 1%, greater than about 10%, greater than about 30%, greater than about 50%, or greater than about 80%.
In some embodiments, the energy transfer operates with a radiation loss η<sub>rad </sub>less that about 10%. In some such embodiments the coupling to loss ratio
<maths id="MATHUS00003" num="00003"><math overflow="scroll"><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><mn>0.1</mn><mo>.</mo></mrow></mrow></math></maths>
In some embodiments, the energy transfer operates with a radiation loss η<sub>rad </sub>less that about 1%. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00004" num="00004"><math overflow="scroll"><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><mn>1.</mn></mrow></math></maths>
In some embodiments, in the presence of a human at distance of more than 3 cm from the surface of either resonant object, the energy transfer operates with a loss to the human η<sub>h </sub>of less than about 1%. In some such embodiments the coupling to loss ratio
<maths id="MATHUS00005" num="00005"><math overflow="scroll"><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><mn>1.</mn></mrow></math></maths>
In some embodiments, in the presence of a human at distance of more than 10 cm from the surface of either resonant object, the energy transfer operates with a loss to the human η<sub>h </sub>of less than about 0.2%. In some such embodiments the coupling to loss ratio
<maths id="MATHUS00006" num="00006"><math overflow="scroll"><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><mn>1.</mn></mrow></math></maths>
In some embodiments, during operation, a device coupled to the first or second resonator structure with a coupling rate Γ<sub>work </sub>receives a usable power P<sub>work </sub>from the resonator structure.
In some embodiments, P<sub>work </sub>is greater than about 0.01 Watt, greater than about 0.1 Watt, greater than about 1 Watt, or greater than about 10 Watt.
In some embodiments, if the device is coupled to the first resonator, then ½≦[(Γ<sub>work</sub>/Γ<sub>1</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦2, or ¼≦[(Γ<sub>work</sub>/Γ<sub>1</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦4, or ⅛≦[(Γ<sub>work</sub>/Γ<sub>1</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦8, and, if the device is coupled to the second resonator, then ½≦[(Γ<sub>work</sub>/Γ<sub>2</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦2, or ¼≦[(Γ<sub>work</sub>/Γ<sub>2</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦4, or ⅛≦[(Γ<sub>work</sub>/Γ<sub>2</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦8.
In some embodiments, the device includes an electrical or electronic device. In some embodiments, the device includes a robot (e.g. a conventional robot or a nanorobot). In some embodiments, the device includes a mobile electronic device (e.g. a telephone, or a cellphone, or a computer, or a laptop computer, or a personal digital assistant (PDA)). In some embodiments, the device includes an electronic device that receives information wirelessly (e.g. a wireless keyboard, or a wireless mouse, or a wireless computer screen, or a wireless television screen). In some embodiments, the device includes a medical device configured to be implanted in a patient (e.g. an artificial organ, or implant configured to deliver medicine). In some embodiments, the device includes a sensor. In some embodiments, the device includes a vehicle (e.g. a transportation vehicle, or an autonomous vehicle).
In some embodiments, the apparatus further includes the device.
In some embodiments, during operation, a power supply coupled to the first or second resonator structure with a coupling rate Γ<sub>supply </sub>drives the resonator structure at a frequency f and supplies power P<sub>total</sub>. In some embodiments, the absolute value of the difference of the angular frequencies ω=2πf and ω<sub>1 </sub>is smaller than the resonant width Γ<sub>1</sub>, and the absolute value of the difference of the angular frequencies ω=2πf and ω<sub>2 </sub>is smaller than the resonant width Γ<sub>2</sub>. In some embodiments, f is about the optimum efficiency frequency.
In some embodiments, if the power supply is coupled to the first resonator, then ½≦[(Γ<sub>supply</sub>/Γ<sub>1</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦2, or ¼≦[(Γ<sub>supply</sub>/Γ<sub>1</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦4, or ⅛≦[(Γ<sub>supply</sub>/Γ<sub>1</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦8, and, if the power supply is coupled to the second resonator, then ½≦[(Γ<sub>supply</sub>/Γ<sub>1</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦2, or ¼≦[(Γ<sub>supply</sub>/Γ<sub>1</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦4, or ⅛≦[(Γ<sub>supply</sub>/Γ<sub>1</sub>)<sup>2</sup>−1]/(κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)})<sup>2</sup>≦8.
In some embodiments, the apparatus further includes the power source.
In some embodiments, the resonant fields are electromagnetic. In some embodiments, f is about 50 GHz or less, about 1 GHz or less, about 100 MHz or less, about 10 MHz or less, about 1 MHz or less, about 100 KHz or less, or about 10 kHz or less. In some embodiments, f is about 50 GHz or greater, about 1 GHz or greater, about 100 MHz or greater, about 10 MHz or greater, about 1 MHz or greater, about 100 kHz or greater, or about 10 kHz or greater. In some embodiments, f is within one of the frequency bands specially assigned for industrial, scientific and medical (ISM) equipment.
In some embodiments, the resonant fields are primarily magnetic in the area outside of the resonant objects. In some such embodiments, the ratio of the average electric field energy to average magnetic filed energy at a distance D<sub>p </sub>from the closest resonant object is less than 0.01, or less than 0.1. In some embodiments, L<sub>R </sub>is the characteristic size of the closest resonant object and D<sub>p</sub>/L<sub>R </sub>is less than 1.5, 3, 5, 7, or 10.
In some embodiments, the resonant fields are acoustic. In some embodiments, one or more of the resonant fields include a whispering gallery mode of one of the resonant structures.
In some embodiments, one of the first and second resonator structures includes a self resonant coil of conducting wire, conducting Litz wire, or conducting ribbon. In some embodiments, both of the first and second resonator structures include self resonant coils of conducting wire, conducting Litz wire, or conducting ribbon. In some embodiments, both of the first and second resonator structures include self resonant coils of conducting wire or conducting Litz wire or conducting ribbon, and Q<sub>1</sub>>300 and Q<sub>2</sub>>300.
In some embodiments, one or more of the self resonant conductive wire coils include a wire of length l and cross section radius a wound into a helical coil of radius r, height h and number of turns N. In some embodiments, N=√{square root over (l<sup>2</sup>−h<sup>2</sup>)}/2πr.
In some embodiments, for each resonant structure r is about 30 cm, h is about 20 cm, a is about 3 mm and N is about 5.25, and, during operation, a power source coupled to the first or second resonator structure drives the resonator structure at a frequency f. In some embodiments, f is about 10.6 MHz. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00007" num="00007"><math overflow="scroll"><mrow><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><mn>40</mn></mrow><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><mn>15</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mn>5</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mn>1.</mn></mrow></mrow></math></maths>
In some such embodiments D/L<sub>R </sub>is as large as about 2, 3, 5, or 8.
In some embodiments, for each resonant structure r is about 30 cm, h is about 20 cm, a is about 1 cm and N is about 4, and, during operation, a power source coupled to the first or second resonator structure drives the resonator structure at a frequency f. In some embodiments, f is about 13.4 MHz. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00008" num="00008"><math overflow="scroll"><mrow><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><mn>70</mn></mrow><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><mn>19</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mn>8</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mn>3.</mn></mrow></mrow></math></maths>
In some such embodiments D/L<sub>R </sub>is as large as about 3, 5, 7, or 10.
In some embodiments, for each resonant structure r is about 10 cm, h is about 3 cm, a is about 2 mm and N is about 6, and, during operation, a power source coupled to the first or second resonator structure drives the resonator structure at a frequency f. In some embodiments, f is about 21.4 MHz. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00009" num="00009"><math overflow="scroll"><mrow><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><mn>59</mn></mrow><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><mn>15</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mn>6</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mn>2.</mn></mrow></mrow></math></maths>
In some such embodiments D/L<sub>R </sub>is as large as about 3, 5, 7, or 10.
In some embodiments, one of the first and second resonator structures includes a capacitively loaded loop or coil of conducting wire, conducting Litz wire, or conducting ribbon. In some embodiments, both of the first and second resonator structures include capacitively loaded loops or coils of conducting wire, conducting Litz wire, or conducting ribbon. In some embodiments, both of the first and second resonator structures include capacitively loaded loops or coils of conducting wire or conducting Litz wire or conducting ribbon, and Q<sub>1</sub>>300 and Q<sub>2</sub>>300.
In some embodiments, the characteristic size L<sub>R </sub>of the resonator structure receiving energy from the other resonator structure is less than about 1 cm and the width of the conducting wire or Litz wire or ribbon of said object is less than about 1 mm, and, during operation, a power source coupled to the first or second resonator structure drives the resonator structure at a frequency f. In some embodiments, f is about 380 MHz. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00010" num="00010"><math overflow="scroll"><mrow><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><mn>14.9</mn></mrow><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><mn>3.2</mn></mrow><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><mn>1.2</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mrow><mn>0.4</mn><mo>.</mo></mrow></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>R </sub>is as large as about 3, about 5, about 7, or about 10.
In some embodiments, the characteristic size of the resonator structure receiving energy from the other resonator structure L<sub>R </sub>is less than about 10 cm and the width of the conducting wire or Litz wire or ribbon of said object is less than about 1 cm, and, during operation, a power source coupled to the first or second resonator structure drives the resonator structure at a frequency f. In some embodiments, f is about 43 MHz. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00011" num="00011"><math overflow="scroll"><mrow><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><mn>15.9</mn></mrow><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><mn>4.3</mn></mrow><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><mn>1.8</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mrow><mn>0.7</mn><mo>.</mo></mrow></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>R </sub>is as large as about 3, about 5, about 7, or about 10.
In some embodiments, the characteristic size L<sub>R </sub>of the resonator structure receiving energy from the other resonator structure is less than about 30 cm and the width of the conducting wire or Litz wire or ribbon of said object is less than about 5 cm, and, during operation, a power source coupled to the first or second resonator structure drives the resonator structure at a frequency f. In some such embodiments, f is about 9 MHz. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00012" num="00012"><math overflow="scroll"><mrow><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><mn>67.4</mn></mrow><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><mn>17.8</mn></mrow><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><mn>7.1</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mrow><mn>2.7</mn><mo>.</mo></mrow></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>R </sub>is as large as about 3, about 5, about 7, or about 10.
In some embodiments, the characteristic size of the resonator structure receiving energy from the other resonator structure L<sub>R </sub>is less than about 30 cm and the width of the conducting wire or Litz wire or ribbon of said object is less than about 5 mm, and, during operation, a power source coupled to the first or second resonator structure drives the resonator structure at a frequency f. In some embodiments, f is about 17 MHz. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00013" num="00013"><math overflow="scroll"><mrow><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><mn>6.3</mn></mrow><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><mn>1.3</mn></mrow><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><mn>0.5</mn><mo>.</mo></mrow></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mrow><mn>0.2</mn><mo>.</mo></mrow></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>R </sub>is as large as about 3, about 5, about 7, or about 10.
In some embodiments, the characteristic size L<sub>R </sub>of the resonator structure receiving energy from the other resonator structure is less than about 1 m, and the width of the conducting wire or Litz wire or ribbon of said object is less than about lcm, and, during operation, a power source coupled to the first or second resonator structure drives the resonator structure at a frequency f. In some embodiments, f is about 5 MHz. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00014" num="00014"><math overflow="scroll"><mrow><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><mn>6.8</mn></mrow><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><mn>1.4</mn></mrow><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><mn>0.5</mn></mrow><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><mn>0.2</mn><mo>.</mo></mrow></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>R </sub>is as large as about 3, about 5, about 7, or about 10.
In some embodiments, during operation, one of the resonator structures receives a usable power P<sub>w </sub>from the other resonator structure, an electrical current I<sub>s </sub>flows in the resonator structure which is transferring energy to the other resonant structure, and the ratio
<maths id="MATHUS00015" num="00015"><math overflow="scroll"><mfrac><msub><mi>I</mi><mi>s</mi></msub><msqrt><msub><mi>P</mi><mi>w</mi></msub></msqrt></mfrac></math></maths>
is less than about 5 Amps/√{square root over (Watts)} or less than about 2 Amps/√{square root over (Watts)}. In some embodiments, during operation, one of the resonator structures receives a usable power P<sub>w </sub>from the other resonator structure, a voltage difference V<sub>s </sub>appears across the capacitive element of the first resonator structure, and the ratio
<maths id="MATHUS00016" num="00016"><math overflow="scroll"><mfrac><msub><mi>V</mi><mi>s</mi></msub><msqrt><msub><mi>P</mi><mi>w</mi></msub></msqrt></mfrac></math></maths>
is less than about 2000 Volts/√{square root over (Watts)} or less than about 4000 Volts/√{square root over (Watts)}.
In some embodiments, one of the first and second resonator structures includes a inductively loaded rod of conducting wire or conducting Litz wire or conducting ribbon. In some embodiments, both of the first and second resonator structures include inductively loaded rods of conducting wire or conducting Litz wire or conducting ribbon. In some embodiments, both of the first and second resonator structures include inductively loaded rods of conducting wire or conducting Litz wire or conducting ribbon, and Q<sub>2</sub>>300 and Q<sub>2</sub>>300.
In some embodiments, the characteristic size of the resonator structure receiving energy from the other resonator structure L<sub>R </sub>is less than about 10 cm and the width of the conducting wire or Litz wire or ribbon of said object is less than about 1 cm, and, during operation, a power source coupled to the first or second resonator structure drives the resonator structure at a frequency f. In some embodiments, f is about 14 MHz. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00017" num="00017"><math overflow="scroll"><mrow><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><mn>32</mn></mrow><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><mn>5.8</mn></mrow><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><mn>2</mn></mrow><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><mn>0.6</mn><mo>.</mo></mrow></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>R </sub>is as large as about 3, about 5, about 7, or about 10.
In some embodiments, the characteristic size L<sub>R </sub>of the resonator structure receiving energy from the other resonator structure is less than about 30 cm and the width of the conducting wire or Litz wire or ribbon of said object is less than about 5 cm, and, during operation, a power source coupled to the first or second resonator structure drives the resonator structure at a frequency f. In some such embodiments, f is about 2.5 MHz. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00018" num="00018"><math overflow="scroll"><mrow><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><mn>105</mn></mrow><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><mn>19</mn></mrow><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><mn>6.6</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mrow><mn>2.2</mn><mo>.</mo></mrow></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>R </sub>is as large as about 3, about 5, about 7, or about 10.
In some embodiments, one of the first and second resonator structures includes a dielectric disk. In some embodiments, both of the first and second resonator structures include dielectric disks. In some embodiments, both of the first and second resonator structures include dielectric disks, and Q<sub>1</sub>>300 and Q<sub>2</sub>>300.
In some embodiments, the characteristic size of the resonator structure receiving energy from the other resonator structure is L<sub>R </sub>and the real part of the permittivity of said resonator structure c is less than about 150. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00019" num="00019"><math overflow="scroll"><mrow><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><mn>42.4</mn></mrow><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><mn>6.5</mn></mrow><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><mn>2.3</mn></mrow><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><mn>0.5</mn><mo>.</mo></mrow></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>R </sub>is as large as about 3, about 5, about 7, or about 10.
In some embodiments, the characteristic size of the resonator structure receiving energy from the other resonator structure is L<sub>R </sub>and the real part of the permittivity of said resonator structure ∈ is less than about 65. In some such embodiments, the coupling to loss ratio
<maths id="MATHUS00020" num="00020"><math overflow="scroll"><mrow><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><mn>30.9</mn></mrow><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><mn>2.3</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>≥</mo><mrow><mn>0.5</mn><mo>.</mo></mrow></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>R </sub>is as large as about 3, about 5, about 7.
In some embodiments, at least one of the first and second resonator structures includes one of: a dielectric material, a metallic material, a metallodielectric object, a plasmonic material, a plasmonodielectric object, a superconducting material.
In some embodiments, at least one of the resonators has a quality factor greater than about 5000, or greater than about 10000.
In some embodiments, the apparatus also includes a third resonator structure configured to transfer energy with one or more of the first and second resonator structures, where the energy transfer between the third resonator structure and the one or more of the first and second resonator structures is mediated by evanescenttail coupling of the resonant field of the one or more of the first and second resonator structures and a resonant field of the third resonator structure.
In some embodiments, the third resonator structure is configured to transfer energy to one or more of the first and second resonator structures.
In some embodiments, the first resonator structure is configured to receive energy from one or more of the first and second resonator structures.
In some embodiments, the first resonator structure is configured to receive energy from one of the first and second resonator structures and transfer energy to the other one of the first and second resonator structures.
Some embodiments include a mechanism for, during operation, maintaining the resonant frequency of one or more of the resonant objects. In some embodiments, the feedback mechanism comprises an oscillator with a fixed frequency and is configured to adjust the resonant frequency of the one or more resonant objects to be about equal to the fixed frequency. In some embodiments, the feedback mechanism is configured to monitor an efficiency of the energy transfer, and adjust the resonant frequency of the one or more resonant objects to maximize the efficiency.
In another aspect, a method of wireless energy transfer is disclosed, which method includes providing a first resonator structure and transferring energy with a second resonator structure over a distance D greater than a characteristic size L<sub>2 </sub>of the second resonator structure. In some embodiments, D is also greater than one or more of: a characteristic size L<sub>1 </sub>of the first resonator structure, a characteristic thickness T<sub>1 </sub>of the first resonator structure, and a characteristic width W<sub>1 </sub>of the first resonator structure. The energy transfer is mediated by evanescenttail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure.
In some embodiments, the first resonator structure is configured to transfer energy to the second resonator structure. In some embodiments, the first resonator structure is configured to receive energy from the second resonator structure.
In some embodiments, the first resonator structure has a resonant angular frequency ω<sub>1</sub>, a Qfactor Q<sub>1</sub>, and a resonance width Γ<sub>1</sub>, the second resonator structure has a resonant angular frequency ω<sub>2</sub>, a Qfactor Q<sub>2</sub>, and a resonance width Γ<sub>2</sub>, and the energy transfer has a rate κ. In some embodiments, the absolute value of the difference of the angular frequencies ω<sub>1 </sub>and ω<sub>2 </sub>is smaller than the broader of the resonant widths Γ<sub>1 </sub>and Γ<sub>2</sub>.
In some embodiments, the coupling to loss ratio
<maths id="MATHUS00021" num="00021"><math overflow="scroll"><mrow><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><mn>0.5</mn></mrow><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><mn>1</mn></mrow><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><mn>2</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>></mo><mn>5.</mn></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>2 </sub>may be as large as 2, as large as 3, as large as 5, as large as 7, or as large as 10.
In another aspect, an apparatus is disclosed for use in wireless information transfer which includes a first resonator structure configured to transfer information by transferring energy with a second resonator structure over a distance D greater than a characteristic size L<sub>2 </sub>of the second resonator structure. In some embodiments, D is also greater than one or more of: a characteristic size L<sub>1 </sub>of the first resonator structure, a characteristic thickness T<sub>1 </sub>of the first resonator structure, and a characteristic width W<sub>1 </sub>of the first resonator structure. The energy transfer is mediated by evanescenttail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure.
In some embodiments, the first resonator structure is configured to transfer energy to the second resonator structure. In some embodiments, the first resonator structure is configured to receive energy from the second resonator structure. In some embodiments the apparatus includes, the second resonator structure.
In some embodiments, the first resonator structure has a resonant angular frequency ω<sub>1</sub>, a Qfactor Q<sub>1</sub>, and a resonance width Γ<sub>1</sub>, the second resonator structure has a resonant angular frequency ω<sub>2</sub>, a Qfactor Q<sub>2</sub>, and a resonance width Γ<sub>2</sub>, and the energy transfer has a rate κ. In some embodiments, the absolute value of the difference of the angular frequencies ω<sub>1 </sub>and ω<sub>2 </sub>is smaller than the broader of the resonant widths Γ<sub>1 </sub>and Γ<sub>2</sub>.
In some embodiments, the coupling to loss ratio
<maths id="MATHUS00022" num="00022"><math overflow="scroll"><mrow><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><mn>0.5</mn></mrow><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><mn>1</mn></mrow><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><mn>2</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>></mo><mn>5.</mn></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>2 </sub>may be as large as 2, as large as 3, as large as 5, as large as 7, or as large as 10.
In another aspect, a method of wireless information transfer is disclosed, which method includes providing a first resonator structure and transferring information by transferring energy with a second resonator structure over a distance D greater than a characteristic size L<sub>2 </sub>of the second resonator structure. In some embodiments, D is also greater than one or more of: a characteristic size L<sub>1 </sub>of the first resonator structure, a characteristic thickness T<sub>1 </sub>of the first resonator structure, and a characteristic width W<sub>1 </sub>of the first resonator structure. The energy transfer is mediated by evanescenttail coupling of a resonant field of the first resonator structure and a resonant field of the second resonator structure.
In some embodiments, the first resonator structure is configured to transfer energy to the second resonator structure. In some embodiments, the first resonator structure is configured to receive energy from the second resonator structure.
In some embodiments, the first resonator structure has a resonant angular frequency ω<sub>1</sub>, a Qfactor Q<sub>1</sub>, and a resonance width Γ<sub>1</sub>, the second resonator structure has a resonant angular frequency ω<sub>2</sub>, a Qfactor Q<sub>2</sub>, and a resonance width Γ<sub>2</sub>, and the energy transfer has a rate κ. In some embodiments, the absolute value of the difference of the angular frequencies ω<sub>1 </sub>and ω<sub>2 </sub>is smaller than the broader of the resonant widths Γ<sub>1 </sub>and Γ<sub>2</sub>.
In some embodiments, the coupling to loss ratio
<maths id="MATHUS00023" num="00023"><math overflow="scroll"><mrow><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><mn>0.5</mn></mrow><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><mn>1</mn></mrow><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><mn>2</mn></mrow><mo>,</mo><mrow><mrow><mi>or</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><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></mrow><mo>></mo><mn>5.</mn></mrow></mrow></math></maths>
In some such embodiments, D/L<sub>2 </sub>may be as large as 2, as large as 3, as large as 5, as large as 7, or as large as 10.
It is to be understood that the characteristic size of an object is equal to the radius of the smallest sphere which can fit around the entire object. The characteristic thickness of an object is, 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 is 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.
The distance D over which the energy transfer between two resonant objects occurs is the distance between the respective centers of the smallest spheres which can fit around the entirety of each object. However, when considering the distance between a human and a resonant object, the distance is to be measured from the outer surface of the human to the outer surface of the sphere.
As described in detail below, nonradiative energy transfer refers to energy transfer effected primarily through the localized near field, and, at most, secondarily through the radiative portion of the field.
It is to be understood that an evanescent tail of a resonant object is the decaying nonradiative portion of a resonant field localized at the object. The decay may take any functional form including, for example, an exponential decay or power law decay.
The optimum efficiency frequency of a wireless energy transfer system is the frequency at which the figure of merit
<maths id="MATHUS00024" num="00024"><math overflow="scroll"><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></math></maths>
is maximized, all other factors held constant.
The resonant width (Γ) refers to the width of an object's resonance due to object's intrinsic losses (e.g. loss to absorption, radiation, etc.).
It is to be understood that a Qfactor (Q) is a factor that compares the time constant for decay of an oscillating system's amplitude to its oscillation period. For a given resonator mode with angular frequency ω and resonant width Γ, the Qfactor Q=ω/2Γ.
The energy transfer rate (κ) refers to the rate of energy transfer from one resonator to another. In the coupled mode description described below it is the coupling constant between the resonators.
It is to be understood that Q<sub>κ</sub>=ω/2κ.
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 to which this invention belongs. In case of conflict with publications, patent applications, patents, and other references mentioned incorporated herein by reference, the present specification, including definitions, will control.
Various embodiments may include any of the above features, alone or in combination. Other features, objects, and advantages of the disclosure will be apparent from the following detailed description.
Other features, objects, and advantages of the disclosure will be apparent from the following detailed description.
BRIEF DESCRIPTION OF THE DRAWINGSFIG. 1 shows a schematic of a wireless energy transfer scheme.
FIG. 2 shows an example of a selfresonant conductingwire coil.
FIG. 3 shows a wireless energy transfer scheme featuring two selfresonant conductingwire coils
FIG. 4 shows an example of a capacitively loaded conductingwire coil, and illustrates the surrounding field.
FIG. 5 shows a wireless energy transfer scheme featuring two capacitively loaded conductingwire coils, and illustrates the surrounding field.
FIG. 6 shows an example of a resonant dielectric disk, and illustrates the surrounding field.
FIG. 7 shows a wireless energy transfer scheme featuring two resonant dielectric disks, and illustrates the surrounding field.
FIGS. 8a and 8b show schematics for frequency controlmechanisms.
FIGS. 9a through 9c illustrate a wireless energy transfer scheme in the presence of various extraneous objects.
FIG. 10 illustrates a circuit model for wireless energy transfer.
FIG. 11 illustrates the efficiency of a wireless energy transfer scheme.
FIG. 12 illustrates parametric dependences of a wireless energy transfer scheme. The figure shows efficiency, total (loaded) device Q, and source and device currents, voltages and radiated powers, normalized to 1 Watt of output working power, as functions of 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 plots the parametric dependences of a wireless energy transfer scheme. Efficiency, total (loaded device Q, and source and device currents, voltages and radiated powers (normalized to 1 Watt of output working power) as functions of frequency and wp for a particular choice of source and device loop dimensions, and number of turns Ns and Nd.
FIG. 14 is a schematic of an experimental system demonstrating wireless energy transfer.
FIGS. 1517. Plot experiment results for the system shown schematically in FIG. 14. FIG. 15 shows a comparison of experimental and theoretical values for κ as a function of the separation between the source and device coils. FIG. 16 shows a comparison of experimental and theoretical values for the parameter 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 κ/Γ due to the ˜5% uncertainty in Q.
DETAILED DESCRIPTIONFIG. 1 shows a schematic that generally describes one embodiment 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 L<sub>1 </sub>and a resonant device object of characteristic size L<sub>2</sub>. Both objects are resonant objects. The source object is connected to a power supply (not shown), and the device object is connected to a power consuming device (e.g. a load resistor, not shown). Energy is provided by the power supply to the source object, transferred wirelessly and nonradiatively from the source object to the device object, and consumed by the power consuming device. The wireless nonradiative energy transfer is performed using the field (e.g. the electromagnetic field or acoustic field) of the system of two resonant objects. For simplicity, in the following we will assume that field is the electromagnetic field.
It is to be understood that while two resonant objects are shown in the embodiment of FIG. 1, and in many of the examples below, other embodiments may feature 3 or more resonant objects. For example, in some embodiments a single source object can transfer energy to multiple device objects. In some embodiments energy may be transferred from a first device to a second, and then from the second device to the third, and so forth.
Initially, we present a theoretical framework for understanding nonradiative wireless energy transfer. Note however that it is to be understood that the scope of the invention is not bound by theory.
Coupled Mode Theory
An appropriate analytical framework for modeling the resonant energyexchange between two resonant objects 1 and 2 is that of “coupledmode theory” (CMT). The field of the system of two resonant objects 1 and 2 is approximated by F(r,t)≈a<sub>1</sub>(t)F<sub>1</sub>(r)+a<sub>2</sub>(t)F<sub>2</sub>(r), where F<sub>1,2</sub>(r) are the eigenmodes of 1 and 2 alone, normalized to unity energy, and the field amplitudes a<sub>1,2</sub>(t) are defined so that a<sub>1,2</sub>(t)<sup>2 </sup>is equal to the energy stored inside the objects 1 and 2 respectively. Then, the field amplitudes can be shown to satisfy, to lowest order:
<maths id="MATHUS00025" num="00025"><math overflow="scroll"><mtable><mtr><mtd><mrow><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>i</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>ω</mi><mn>1</mn></msub><mo></mo><mrow><mi>i</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><mi>i</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>κ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>a</mi><mn>2</mn></msub></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo></mo><msub><mi>a</mi><mn>2</mn></msub></mrow><mrow><mo></mo><mi>t</mi></mrow></mfrac><mo>=</mo><mrow><mrow><mrow><mo></mo><mrow><mi>i</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>ω</mi><mn>2</mn></msub><mo></mo><mrow><mi>i</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><msub><mi>a</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><mi>i</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>κ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>a</mi><mn>1</mn></msub></mrow></mrow></mrow></mtd></mtr></mtable><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where ω<sub>1,2 </sub>are the individual angular eigenfrequencies of the eigenmodes, Γ<sub>1,2 </sub>are the resonance widths due to the objects' intrinsic (absorption, radiation etc.) losses, and κ is the coupling coefficient. Eqs. (1) show that at exact resonance (ω<sub>1</sub>=ω<sub>2 </sub>and Γ<sub>1</sub>=Γ<sub>2</sub>), the eigenmodes of the combined system are split by 2κ; the energy exchange between the two objects takes place in time ˜π/2κ and is nearly perfect, apart for losses, which are minimal when the coupling rate is much faster than all loss rates (κ<img id="CUSTOMCHARACTER00003" he="2.12mm" wi="1.44mm" file="US20110140544A120110616P00003.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>Γ<sub>1,2</sub>). The coupling to loss ratio κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)} serves as a figureofmerit in evaluating a system used for wireless energytransfer, along with the distance over which this ratio can be achieved. The regime κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)}>>1 is called “strongcoupling” regime.
In some embodiments, the energytransfer application preferably uses resonant modes of high Q=ω/2Γ, corresponding to low (i.e. slow) intrinsicloss rates Γ. This condition may be satisfied where the coupling is implemented using, not the lossy radiative farfield, but the evanescent (nonlossy) stationary nearfield.
To implement an energytransfer scheme, usually finite objects, namely ones that are topologically surrounded everywhere by air, are more appropriate. Unfortunately, objects of finite extent cannot support electromagnetic states that are exponentially decaying in all directions in air, since, from Maxwell's Equations in free space: {right arrow over (k)}<sup>2</sup>=ω<sup>2</sup>/c<sup>2 </sup>where {right arrow over (k)} is the wave vector, ω the angular frequency, and c the speed of light. Because of this, one can show that they cannot support states of infinite Q. 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.
Furthermore, in some embodiments, small Q<sub>κ</sub>=ω/2κ corresponding to strong (i.e. fast) coupling rate κ is preferred over distances larger than the characteristic sizes of the objects. Therefore, since the extent of the nearfield into the area surrounding a finitesized resonant object is set typically by the wavelength, in some embodiments, this midrange nonradiative coupling can be achieved using resonant objects of subwavelength size, and thus significantly longer evanescent fieldtails. As will be seen in examples later on, such subwavelength resonances can often be accompanied with a high Q, so this will typically be the appropriate choice for the possiblymobile resonant deviceobject. Note, though, that in some embodiments, the resonant sourceobject will be immobile and thus less restricted in its allowed geometry and size, which can be therefore chosen large enough that the nearfield extent is not limited by the wavelength. 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, and can have nearly infinite Q.
In the following, we describe several examples of systems suitable for energy transfer of the type described above. We will demonstrate how to compute the CMT parameters ω<sub>1,2</sub>, Q<sub>1,2 </sub>and Q<sub>κ</sub> described above and how to choose these parameters for particular embodiments in order to produce a desirable figureofmerit κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)}√{square root over (Q<sub>1</sub>Q<sub>2</sub>)}/Q<sub>κ</sub>. In particular, this figure of merit is typically maximized when ω<sub>1,2 </sub>are tuned to a particular angular frequency {tilde over (ω)}, thus, if Γ is half the angularfrequency width for which √{square root over (Q<sub>1</sub>Q<sub>2</sub>)}/Q<sub>κ</sub> is above half its maximum value at {tilde over (ω)}, the angular eigenfrequencies ω<sub>1,2 </sub>should typically be tuned to be close to {tilde over (ω)} to within the width {tilde over (Γ)}.
In addition, as described below, Q<sub>1,2 </sub>can sometimes be limited not from intrinsic loss mechanisms but from external perturbations. In those cases, producing a desirable figureofmerit translates to reducing Q<sub>κ</sub> (i.e. increasing the coupling). Accordingly we will demonstrate how, for particular embodiments, to reduce Q<sub>κ</sub>.
SelfResonant Conducting Coils
In some embodiments, one or more of the resonant objects are selfresonant conducting coils. Referring to FIG. 2, 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 exchangd 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 U 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="MATHUS00026" num="00026"><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{(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 U inside its resonant mode:
<maths id="MATHUS00027" num="00027"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mrow><mi>U</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><msubsup><mi>πI</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>2</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mrow><mrow><mi>U</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><msub><mi>πɛ</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>3</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 are given by the common 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="MATHUS00028" num="00028"><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><mn>1</mn><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>4</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>5</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where c=1/√{square root over (μ<sub>0</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/σδ=√{square root over (μ<sub>o</sub>ω/2σ)} with σ the conductivity of the conductor and δ the skin depth at the frequency ω,
<maths id="MATHUS00029" num="00029"><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=∫dxrρ<sub>l</sub>(x) is the electricdipole moment of the coil and
<maths id="MATHUS00030" num="00030"><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="0.8em" height="0.8ex"/></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. (5), 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 are given by Q<sup>abs</sup>=Z/R<sub>abs </sub>and Q<sup>rad</sup>=Z/R<sub>rad </sub>respectively.
From Eq. (2)(5) 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>1</sub>(x)=ρ<sub>o </sub>sin(πx/l), and thus q<sub>o</sub>=∫<sub>0</sub><sup>1/2</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. (2) and (3); 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. (4) and (5) using
<maths id="MATHUS00031" num="00031"><math overflow="scroll"><mrow><mrow><msubsup><mi>I</mi><mi>rms</mi><mn>2</mn></msubsup><mo>=</mo><mrow><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><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></mrow><mo>,</mo><mrow><mrow><mo></mo><mi>p</mi><mo></mo></mrow><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><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></mrow></mrow></mrow></math></maths><maths id="MATHUS000312" num="00031.2"><math overflow="scroll"><mi>and</mi></math></maths><maths id="MATHUS000313" num="00031.3"><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><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><msup><mi>N</mi><mn>2</mn></msup></mrow><mo></mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo></mo><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><msup><mi>N</mi><mn>2</mn></msup></mrow><mo></mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mrow><mn>16</mn><mo></mo><msup><mi>N</mi><mn>4</mn></msup></mrow><mo></mo><mrow><mn>8</mn><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 may be calculated.
The results for two particular embodiments 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</sub><sup>abs</sup>≧1000 Q<sub>a</sub><sup>rad</sup>≧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="70pt" align="left"/><colspec colname="2" colwidth="28pt" 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="42pt" 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</sub><sup>rad</sup></entry><entry>Q<sub>s</sub><sup>abs</sup></entry><entry>Q<sub>s </sub>= ω<sub>s</sub>/2Γ<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="70pt" align="left"/><colspec colname="2" colwidth="28pt" 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="42pt" align="center"/><tbody valign="top"><row><entry>r = 30 cm, h = 20 cm,</entry><entry>74.7</entry><entry>13.39</entry><entry>4164</entry><entry>8170</entry><entry>2758</entry></row><row><entry>a = 1 cm, N = 4</entry></row><row><entry>r = 10 cm, h = 3 cm,</entry><entry>140</entry><entry>21.38</entry><entry>43919</entry><entry>3968</entry><entry>3639</entry></row><row><entry>a = 2 mm, N = 6</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
Referring to FIG. 3, in some embodiments, 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>0</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="MATHUS00032" num="00032"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mrow><mi>U</mi><mo>≡</mo><mrow><msub><mi>U</mi><mn>1</mn></msub><mo>+</mo><msub><mi>U</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><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><msub><mi>πɛ</mi><mi>o</mi></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><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></mrow></mtd><mtd><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where
<maths id="MATHUS00033" num="00033"><math overflow="scroll"><mrow><mrow><msub><mi>U</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>U</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></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 coefficient is given by κ=ω√{square root over (C<sub>1</sub>C<sub>2</sub>)}/2M<sub>C</sub>+ωM<sub>L</sub>/2√{square root over (L<sub>1</sub>L<sub>2</sub>)}<img id="CUSTOMCHARACTER00004" he="2.46mm" wi="3.13mm" file="US20110140544A120110616P00004.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>Q<sub>κ</sub><sup>−1</sup>=(M<sub>C</sub>/√{square root over (C<sub>1</sub>C<sub>2</sub>)})<sup>−1</sup>+(√{square root over (L<sub>1</sub>L<sub>2</sub>)}/M<sub>L</sub>)<sup>−</sup>1.
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. (6) 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 Q<sub>κs </sub>is also determined.
<tables id="TABLEUS00002" num="00002"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="5"><colspec colname="1" colwidth="84pt" align="left"/><colspec colname="2" colwidth="21pt" align="center"/><colspec colname="3" colwidth="42pt" align="center"/><colspec colname="4" colwidth="42pt" align="center"/><colspec colname="5" colwidth="28pt" align="center"/><thead><row><entry namest="1" nameend="5" rowsep="1">TABLE 2</entry></row><row><entry namest="1" nameend="5" align="center" rowsep="1"/></row><row><entry>pair of coils</entry><entry>D/r</entry><entry>Q = ω/2Γ</entry><entry>Q<sub>κ </sub>= ω/2κ</entry><entry>κ/Γ</entry></row><row><entry namest="1" nameend="5" align="center" rowsep="1"/></row></thead><tbody valign="top"><row><entry/></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="5"><colspec colname="1" colwidth="84pt" align="left"/><colspec colname="2" colwidth="21pt" align="char" char="."/><colspec colname="3" colwidth="42pt" align="center"/><colspec colname="4" colwidth="42pt" align="char" char="."/><colspec colname="5" colwidth="28pt" align="char" char="."/><tbody valign="top"><row><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>a = 1 cm, N = 4</entry><entry>5</entry><entry>2758</entry><entry>139.4</entry><entry>19.8</entry></row><row><entry>λ/r ≈ 75</entry><entry>7</entry><entry>2758</entry><entry>333.0</entry><entry>8.3</entry></row><row><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>r = 10 cm, h = 3 cm,</entry><entry>3</entry><entry>3639</entry><entry>61.4</entry><entry>59.3</entry></row><row><entry>a = 2 mm, N = 6</entry><entry>5</entry><entry>3639</entry><entry>232.5</entry><entry>15.7</entry></row><row><entry>λ/r ≈ 140</entry><entry>7</entry><entry>3639</entry><entry>587.5</entry><entry>6.2</entry></row><row><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 namest="1" nameend="5" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
Referring to Table 2, relevant parameters are shown for exemplary embodiments 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=10−3 the expected couplingtoloss ratios are in the range κ/Γ˜270.
CapacitivelyLoaded Conducting Loops or Coils
In some embodiments, one or more of the resonant objects are capacitivelyloaded conducting loops or coils. Referring to FIG. 4 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 ∈, and everything is surrounded by air (as shown, N=1 and h=0). The plates have a capacitance C<sub>p</sub>=∈<sub>0</sub>∈A/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 modifies significantly the current distribution inside the wire and therefore the total effective inductance L and total effective capacitance C of the coil are 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 is accumulated at the plates of the external loading capacitor, the charge distribution ρ inside the wire is reduced, so C<C<sub>s</sub>, and thus, from the charge conservation equation, the current distribution j flattens out, so L>L<sub>s</sub>. The resonant frequency for this system is ω=1/√{square root over (L(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="CUSTOMCHARACTER00005" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>C→C<sub>s</sub><img id="CUSTOMCHARACTER00006" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" 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. (2). In the case h=0 and N integer, the integral in Eq. (2) can actually be computed analytically, giving the formula L=μ<sub>o</sub>r[ln(8r/a)−2]N<sup>2</sup>. Explicit analytical formulas are again available for R from Eq. (4) and (5), 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<sup>abs</sup>=ωL/R<sub>abs </sub>and Q<sup>rad</sup>=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="CUSTOMCHARACTER00007" he="2.46mm" wi="3.13mm" file="US20110140544A120110616P00004.TIF" alt="customcharacter" 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 embodiments, 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="MATHUS00034" num="00034"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mover><mi>ω</mi><mo>~</mo></mover><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><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><mi>N</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msup><mi>r</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>7</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
reaching the value
<maths id="MATHUS00035" num="00035"><math overflow="scroll"><mtable><mtr><mtd><mrow><mover><mi>Q</mi><mo>~</mo></mover><mo>=</mo><mrow><mfrac><mn>6</mn><mrow><mn>7</mn><mo></mo><mi>π</mi></mrow></mfrac><mo></mo><mrow><msup><mrow><mo>(</mo><mrow><mn>2</mn><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><mfrac><mrow><mn>8</mn><mo></mo><mi>r</mi></mrow><mi>a</mi></mfrac><mo>)</mo></mrow><mo></mo><mn>2</mn></mrow><mo>]</mo></mrow><mo>.</mo></mrow></mrow></mrow></mrow></mtd><mtd><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
At lower frequencies it is dominated by ohmic loss and at higher frequencies by radiation. Note, however, that the formulas above are accurate as long as {tilde over (ω)}<<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 {tilde over (ω)}, {tilde over (Q)} 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 embodiments 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. (7) 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 (λ<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 for conducting material copper (σ=5.998·10<sup>7</sup>S/m) was used. (The specific parameters of the plot in FIG. 4 are highlighted with bold in the table.) The two methods (analytical and computational) are in very good agreement and show that, in some embodiments, the optimal frequency is in the lowMHz microwave range and the expected quality factors are Q<sup>abs</sup>≧1000 and Q<sup>rad</sup>≧10000.
<tables id="TABLEUS00003" num="00003"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="6"><colspec colname="1" colwidth="70pt" align="left"/><colspec colname="2" colwidth="28pt" align="center"/><colspec colname="3" colwidth="28pt" align="center"/><colspec colname="4" colwidth="28pt" align="center"/><colspec colname="5" colwidth="28pt" align="center"/><colspec colname="6" colwidth="35pt" 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 (MHz)</entry><entry>Q<sup>rad</sup></entry><entry>Q<sup>abs</sup></entry><entry>Q = ω/2Γ</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="70pt" align="left"/><colspec colname="2" colwidth="28pt" align="char" char="."/><colspec colname="3" colwidth="28pt" align="center"/><colspec colname="4" colwidth="28pt" align="center"/><colspec colname="5" colwidth="28pt" align="center"/><colspec colname="6" colwidth="35pt" align="center"/><tbody valign="top"><row><entry>r = 30 cm, a = 2 cm</entry><entry>111.4</entry><entry>8.976</entry><entry>29546</entry><entry>4886</entry><entry>4193</entry></row><row><entry>ε = 10,</entry><entry>(112.4)</entry><entry>(8.897)</entry><entry>(30512)</entry><entry>(5117)</entry><entry>(4381)</entry></row><row><entry>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</entry><entry>43.04</entry><entry>10702</entry><entry>1545</entry><entry>1350</entry></row><row><entry>ε = 10,</entry><entry>(70.4)</entry><entry>(42.61)</entry><entry>(10727)</entry><entry>(1604)</entry><entry>(1395)</entry></row><row><entry>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. 5, in some embodiments, 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. (6) by using constant current distributions in the case ω<<ω<sub>s</sub>. In the case h=0, the coupling is only magnetic and again we have an analytical formula, which, in the quasistatic limit r<img id="CUSTOMCHARACTER00008" he="2.12mm" wi="1.44mm" file="US20110140544A120110616P00006.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>D<img id="CUSTOMCHARACTER00009" he="2.12mm" wi="1.44mm" file="US20110140544A120110616P00006.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>λ and for the relative orientation shown in FIG. 4, 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 Q<sub>κ</sub>∝(D/√{square root over (r<sub>1</sub>r<sub>2</sub>)})<sup>3 </sup>is 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="MATHUS00036" num="00036"><math overflow="scroll"><mtable><mtr><mtd><mrow><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><mfrac><msqrt><mrow><msub><mi>Q</mi><mn>1</mn></msub><mo></mo><msub><mi>Q</mi><mn>2</mn></msub></mrow></msqrt><msub><mi>Q</mi><mi>κ</mi></msub></mfrac><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><msup><mrow><mo>(</mo><mrow><mrow><mrow><msqrt><mfrac><msub><mi>πη</mi><mi>o</mi></msub><mi>λσ</mi></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><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>2</mn></msup><mo></mo><msubsup><mi>N</mi><mi>j</mi><mn>2</mn></msubsup></mrow></mrow><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>9</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. (9) it can be seen that the optimal frequency {tilde over (ω)}, where the figureofmerit is maximized to the value <o ostyle="single">(√{square root over (Q<sub>1</sub>Q<sub>2</sub>)}/Q<sub>κ</sub>)</o>, is that where √{square root over (Q<sub>1</sub>Q<sub>2</sub>)} is maximized, since Q<sub>κ</sub> does not depend on frequency (at least for the distances D<img id="CUSTOMCHARACTER00010" he="2.46mm" wi="1.44mm" file="US20110140544A120110616P00002.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>λ of interest for which the quasistatic approximation is still valid). Therefore, the optimal frequency is independent of the distance D between the two coils and lies between the two frequencies where the singlecoil Q<sub>1 </sub>and Q<sub>2 </sub>peak. For same coils, it is given by Eq. (7) and then the figureofmerit Eq. (9) becomes
<maths id="MATHUS00037" num="00037"><math overflow="scroll"><mtable><mtr><mtd><mrow><mover><mrow><mo>(</mo><mfrac><mi>κ</mi><mi>Γ</mi></mfrac><mo>)</mo></mrow><mo>~</mo></mover><mo>=</mo><mrow><mfrac><mover><mi>Q</mi><mo>~</mo></mover><msub><mi>Q</mi><mi>κ</mi></msub></mfrac><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><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>10</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
Typically, one should tune the capacitivelyloaded conducting loops or coils, so that their angular eigenfrequencies are close to {tilde over (ω)} within {tilde over (σ)}, which is half the angular frequency width for which √{square root over (Q<sub>1</sub>Q<sub>2</sub>)}/Q<sub>κ</sub>> <o ostyle="single">(√{square root over (Q<sub>1</sub>Q<sub>2</sub>)}/Q<sub>78 </sub>)</o>/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 κ/Γ as a function of the coupling distance D, for the two cases. Note that the average numerical Γ<sup>rad </sup>shown are again slightly different from the singleloop value of FIG. 3, analytical results for Γ<sup>rad </sup>are not shown but the singleloop value is used. (The specific parameters corresponding to the plot in FIG. 5 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. (6). Indeed the accuracy can be confirmed by their agreement with the computational FEFD modesolver simulations, which give κ through the frequency splitting (=2κ) of the two normal modes of the combined system. The results show that for medium distances D/r=10−3 the expected couplingtoloss ratios are in the range κ/Γ˜0.550.
<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>ε = 10, A = 138 cm<sup>2</sup>, d = 4 mm</entry><entry>5</entry><entry>29577</entry><entry>4194</entry><entry>235 (248)</entry><entry>17.8 (17.6)</entry></row><row><entry>λ/r ≈ 112</entry><entry>7</entry><entry>29128</entry><entry>4185</entry><entry>589 (646)</entry><entry>7.1 (6.8)</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>≈ 1546</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>Optimization of √{square root over (Q<sub>1</sub>Q<sub>2</sub>)}/Q<sub>κ</sub>
In some embodiments, the results above can be used to increase or optimize the performance of a wireless energy transfer system which employs capacitivelyloaded coils. For example, the scaling of Eq. (10) with the different system parameters one sees that to maximize the system figureofmerit κ/Γ one can, for example:
 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 embodiments, this action is 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 a ribbon instead of a circular wire.
 For fixed desired distance D of energy transfer, increase the radius of the loop r. In typical embodiments, this action is limited by physical size considerations.
 For fixed desired distance vs. loopsize ratio D/r, decrease the radius of the loop r. In typical embodiments, this action is limited by physical size considerations.
 Increase the number of turns N. (Even though Eq. (10) 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 embodiments, this action is limited by physical size and possible voltage considerations, as will be discussed in following sections.
 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 embodiments, 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) should be avoided.
 Finally, note that the height of the coil h is another available design parameter, which has an impact to the performance similar to that of its radius r, and thus the design rules are 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 which one should use when designing as system two same singleturn loops with a given radius in order to achieve a specific performance in terms of κ/Γ at a given D/r between them, when the material is copper (σ=5.998·10<sup>7</sup>S/m):
<FORM>D/r=5, κ/Γ≧10, r=30 cm<img id="CUSTOMCHARACTER00011" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>a≧9 mm</FORM>
<FORM>D/r=5, κ/Γ≧10, r=5 cm<img id="CUSTOMCHARACTER00012" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>a≧3.7 mm</FORM>
<FORM>D/r=5, κ/Γ≧20, r=30 cm<img id="CUSTOMCHARACTER00013" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>a≧20 mm</FORM>
<FORM>D/r=5, κ/Γ≧20, r=5 cm<img id="CUSTOMCHARACTER00014" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>a≧8.3 mm</FORM>
<FORM>D/r=10, κ/Γ≧1, r=30 cm<img id="CUSTOMCHARACTER00015" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>a≧7 mm</FORM>
<FORM>D/r=10, κ/Γ≧1, r=5 cm<img id="CUSTOMCHARACTER00016" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>a≧2.8 mm</FORM>
<FORM>D/r=10, κ/Γ≧3, r=30 cm<img id="CUSTOMCHARACTER00017" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>a≧25 mm</FORM>
<FORM>D/r=10, κ/Γ≧3, r=5 cm<img id="CUSTOMCHARACTER00018" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>a≧10 mm</FORM>
Similar analysis can be done for the case of two dissimilar loops. For example, in some embodiments, the device under consideration is very specific (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>) are very restricted. However, in some such embodiments, the restrictions on the source object (r<sub>s</sub>,h<sub>s</sub>,a<sub>s</sub>,N<sub>s</sub>) are much less, since the source can, for example, be placed under the floor or on the ceiling. In such cases, the desired distance is often well defined, based on the application (e.g. D˜1 m for charging a laptop on a table wirelessly from the floor). 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 the desired system performance in terms of 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/r=1.5 m, κ/√{square root over (Γ<sub>s</sub>Γ<sub>d</sub>)}≧15, r<sub>d</sub>=30 cm, a<sub>d</sub>=6 mm<img id="CUSTOMCHARACTER00019" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.158 m, a<sub>s</sub>≧5 mm</FORM>
<FORM>D/r=1.5 m, κ/√{square root over (Γ<sub>s</sub>Γ<sub>d</sub>)}≧30, r<sub>d</sub>=30 cm, a<sub>d</sub>=6 mm<img id="CUSTOMCHARACTER00020" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.15 m, a<sub>s</sub>≧33 mm</FORM>
<FORM>D/r=1.5 m, κ/√{square root over (Γ<sub>s</sub>Γ<sub>d</sub>)}≧1, r<sub>d</sub>=5 cm, a<sub>d</sub>=4 mm<img id="CUSTOMCHARACTER00021" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.119 m, a<sub>s</sub>≧7 mm</FORM>
<FORM>D/r=1.5 m, κ/√{square root over (Γ<sub>s</sub>Γ<sub>d</sub>)}≧2, r<sub>d</sub>=5 cm, a<sub>d</sub>=4 mm<img id="CUSTOMCHARACTER00022" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.119 m, a<sub>s</sub>≧52 mm</FORM>
<FORM>D/r=2 m, κ/√{square root over (Γ<sub>s</sub>Γ<sub>d</sub>)}≧10, r<sub>d</sub>=30 cm, a<sub>d</sub>=6 mm<img id="CUSTOMCHARACTER00023" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.518 m, a<sub>s</sub>≧7 mm</FORM>
<FORM>D/r=2 m, κ/√{square root over (Γ<sub>s</sub>Γ<sub>d</sub>)}≧20, r<sub>d</sub>=30 cm, a<sub>d</sub>=6 mm<img id="CUSTOMCHARACTER00024" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.518 m, a<sub>s</sub>≧50 mm</FORM>
<FORM>D/r=2 m, κ/√{square root over (Γ<sub>s</sub>Γ<sub>d</sub>)}≧0.5, r<sub>d</sub>=5 cm, a<sub>d</sub>=4 mm<img id="CUSTOMCHARACTER00025" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.491 m, a<sub>s</sub>≧5 mm</FORM>
<FORM>D/r=2 m, κ/√{square root over (Γ<sub>s</sub>Γ<sub>d</sub>)}≧0.5, r<sub>d</sub>=5 cm, a<sub>d</sub>=4 mm<img id="CUSTOMCHARACTER00026" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>r<sub>s</sub>=1.491 m, a<sub>s</sub>≧36 mm</FORM>
Optimization of Q<sub>κ</sub>
As will be described below, in some embodiments the quality factor Q of the resonant objects is limited from external perturbations and thus varying the coil parameters cannot lead to improvement in Q. In such cases, one may opt to increase the coupling to loss ratio figureofmerit by decreasing Q<sub>κ</sub> (i.e. increasing the coupling). The coupling does not depend on the frequency and the number of turns. Therefore, the remaining degrees of freedom are:
 Increase the wire radii a<sub>l </sub>and a<sub>2</sub>. In typical embodiments, this action is 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 embodiments, this action is 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 embodiments, this action is limited by physical size considerations.
 Adjust the alignment and orientation between the two coils. In typical embodiments, 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) 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 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 are similar.
Further practical considerations apart from efficiency, e.g. physical size limitations, will be discussed in detail below.
It is also important to appreciate the difference between the above described resonantcoupling inductive scheme and the wellknown nonresonant inductive scheme for energy transfer. Using CMT it is easy to show that, keeping the geometry and the energy stored at the source fixed, the resonant inductive mechanism allows for ˜Q<sup>2 </sup>(˜10<sup>6</sup>) times more power delivered for work at the device than the traditional nonresonant mechanism. This is why only closerange contactless mediumpower (˜W) transfer is possible with the latter, while with resonance either closerange but largepower (˜kW) transfer is allowed or, as currently proposed, if one also ensures operation in the stronglycoupled regime, mediumrange and mediumpower transfer is possible. Capacitivelyloaded conducting loops are currently used as resonant antennas (for example in cell phones), but those operate in the farfield regime with D/r<img id="CUSTOMCHARACTER00027" he="2.12mm" wi="1.44mm" file="US20110140544A120110616P00003.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>1,r/λ˜1, and the radiation Q's are intentionally designed to be small to make the antenna efficient, so they are not appropriate for energy transfer.
InductivelyLoaded Conducting Rods
A straight conducting rod of length 2h and crosssectional radius a has distributed capacitance and distributed inductance, and therefore it supports 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 (2) and (3). With these definitions, the resonant angular frequency and the effective impedance are given again by the common formulas ω=1/√{square root over (LC)} and Z=√{square root over (L/C)} 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 would 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 embodiments, one or more of the resonant objects are inductivelyloaded conducting rods. 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 are 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 modifies significantly the current distribution inside the wire and therefore the total effective inductance L and total effective capacitance C of the rod are 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 is running inside the coil of the external loading inductor, the current distribution j inside the rod is reduced, so L<L<sub>s</sub>, and thus, from the charge conservation equation, the linear charge distribution ρ<sub>1 </sub>flattens 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 is 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="CUSTOMCHARACTER00028" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>L→L<sub>s</sub><img id="CUSTOMCHARACTER00029" he="2.46mm" wi="2.79mm" file="US20110140544A120110616P00005.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>ω→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 required. Instead, we will analyze only a special case, where a reasonable guess 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 is positive constant on one half of the rod and equally negative constant on the other side of the rod. This allows us now to compute numerically C from Eq. (3). In this case, the integral in Eq. (3) 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. (4) and (5), 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<sup>abs</sup>1/ωCR<sub>abs </sub>and Q<sup>rad</sup>=1/ωCR<sub>rad</sub>. At the end of the calculations, the validity of the assumption of triangular current profile is confirmed by checking that indeed the condition L<sub>c</sub>>>L<sub>s</sub><img id="CUSTOMCHARACTER00030" he="2.46mm" wi="3.13mm" file="US20110140544A120110616P00004.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>ω<<ω<sub>s </sub>is satisfied. This condition is 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 depends on the particular design of the inductor. In some embodiments, the inductor is made of a Brooks coil, which is the coil geometry which, for fixed wire length, demonstrates 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<sub>Bc </sub>(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>/8 a<sub>Bc</sub><sup>4 </sup>and its resistance
<maths id="MATHUS00038" num="00038"><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>ωσ</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)} is 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 is Q<sub>c</sub>≈0.169 μ<sub>o</sub>σr<sub>Bc</sub><sup>2</sup>ω/√{square root over (1+ω<sup>2</sup>μ<sub>o</sub>σ√{square root over (2.0285 μ<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=1ωC(R<sub>c</sub>+R<sub>abs</sub>+R<sub>rad</sub>) becomes highest at some optimal frequency {tilde over (ω)}, reaching the value {tilde over (Q)}, both determined by the loadinginductor specific design. (For example, for the Brookscoil procedure described above, at the optimal frequency {tilde over (Q)}≈Q<sub>c</sub>≈0.8(μ<sub>o</sub>σ<sup>2</sup>r<sub>Bc</sub><sup>3</sup>/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 {tilde over (ω)}<<ω<sub>s </sub>and, as explained above, this is easy to design for by using a large inductance.
The results of the above analysis for two embodiments, 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 {tilde over (ω)} 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 subwavelengthloop resonant modes. Note that for conducting material copper (σ=5.998·10<sup>7 </sup>S/m) was used. The results show that, in some embodiments, the optimal frequency is in the lowMHz microwave range and the expected quality factors are Q<sup>abs</sup>≧1000 and Q<sup>rad</sup>≧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="49pt" 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="35pt" 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<sup>rad</sup></entry><entry>Q<sup>abs</sup></entry><entry>Q = ω/2Γ</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="49pt" 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="35pt" 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,</entry></row><row><entry>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</entry></row><row><entry>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 namest="1" nameend="6" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
In some embodiments, energy is 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. (6) by using triangular current distributions in the case ω<<ω<sub>s</sub>. In this case, the coupling is only electric and again we have an analytical formula, which, in the quasistatic limit h<img id="CUSTOMCHARACTER00031" he="2.46mm" wi="1.44mm" file="US20110140544A120110616P00002.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>D<img id="CUSTOMCHARACTER00032" he="2.46mm" wi="1.44mm" file="US20110140544A120110616P00002.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>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 Q<sub>κ</sub>∝(D/√{square root over (h<sub>1</sub>h<sub>2</sub>)})<sup>3 </sup>is independent of the frequency ω. Consequently, one can get the resultant coupling figureofmerit of interest
<maths id="MATHUS00039" num="00039"><math overflow="scroll"><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><mfrac><msqrt><mrow><msub><mi>Q</mi><mn>1</mn></msub><mo></mo><msub><mi>Q</mi><mn>2</mn></msub></mrow></msqrt><msub><mi>Q</mi><mi>κ</mi></msub></mfrac><mo>.</mo></mrow></mrow></math></maths>
It can be seen that the optimal frequency {tilde over (ω)}, where the figureofmerit is maximized to the value <o ostyle="single">(√{square root over (Q<sub>1</sub>Q<sub>2</sub>)}/Q<sub>κ</sub>), </o>is that where √{square root over (Q<sub>1</sub>Q<sub>2</sub>)} is maximized, since Q<sub>κ</sub> does not depend on frequency (at least for the distances D<img id="CUSTOMCHARACTER00033" he="2.46mm" wi="1.44mm" file="US20110140544A120110616P00002.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>λ of interest for which the quasistatic approximation is still valid). Therefore, the optimal frequency is independent of the distance D between the two rods and lies between the two frequencies where the singlerod Q<sub>1 </sub>and Q<sub>2 </sub>peak. Typically, one should tune the inductivelyloaded conducting rods, so that their angular eigenfrequencies are close to {tilde over (ω)} within {tilde over (Γ)}, which is half the angular frequency width for which √{square root over (Q<sub>1</sub>Q<sub>2</sub>)}/Q<sub>κ</sub>> <o ostyle="single">(√{square root over (Q<sub>1</sub>Q<sub>2</sub>)}/Q<sub>κ</sub>)</o>/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 κ/Γ as a function of the coupling distance D, for the two cases. Note that for Γ<sup>rad </sup>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. (6). The results show that for medium distances D/h=10−3 the expected couplingtoloss ratios are in the range κ/Γ˜0.5100.
<tables id="TABLEUS00006" num="00006"><table frame="none" colsep="0" rowsep="0"><tgroup align="left" colsep="0" rowsep="0" cols="5"><colspec colname="offset" colwidth="14pt" align="left"/><colspec colname="1" colwidth="70pt" align="left"/><colspec colname="2" colwidth="35pt" align="center"/><colspec colname="3" colwidth="42pt" align="center"/><colspec colname="4" colwidth="56pt" align="center"/><thead><row><entry/><entry namest="offset" nameend="4" rowsep="1">TABLE 6</entry></row><row><entry/><entry namest="offset" nameend="4" align="center" rowsep="1"/></row><row><entry/><entry>pair of rods</entry><entry>D/h</entry><entry>Q<sub>κ </sub>= ω/2κ</entry><entry>κ/Γ</entry></row><row><entry/><entry namest="offset" nameend="4" align="center" rowsep="1"/></row></thead><tbody valign="top"><row><entry/></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="5"><colspec colname="offset" colwidth="14pt" align="left"/><colspec colname="1" colwidth="70pt" align="left"/><colspec colname="2" colwidth="35pt" align="char" char="."/><colspec colname="3" colwidth="42pt" align="center"/><colspec colname="4" colwidth="56pt" align="center"/><tbody valign="top"><row><entry/><entry>h = 30 cm, a = 2 cm</entry><entry>3</entry><entry> (70.3)</entry><entry>(105.0) </entry></row><row><entry/><entry>μ = 1, r<sub>Bc </sub>= 2 cm,</entry><entry>5</entry><entry> (389)</entry><entry>(19.0) </entry></row><row><entry/><entry>a<sub>Bc </sub>= 0.88 mm,</entry><entry>7</entry><entry>(1115)</entry><entry>(6.62)</entry></row><row><entry/><entry>N<sub>Bc </sub>= 129</entry><entry>10</entry><entry>(3321)</entry><entry>(2.22)</entry></row><row><entry/><entry>λ/h ≈ 404</entry></row><row><entry/><entry>Q ≈ 7380</entry></row><row><entry/><entry>h = 10 cm, a = 2 mm</entry><entry>3</entry><entry> (120)</entry><entry>(32.4) </entry></row><row><entry/><entry>μ = 1, r<sub>Bc </sub>= 5 mm,</entry><entry>5</entry><entry> (664)</entry><entry>(5.85)</entry></row><row><entry/><entry>a<sub>Bc </sub>= 0.25 mm,</entry><entry>7</entry><entry>(1900)</entry><entry>(2.05)</entry></row><row><entry/><entry>N<sub>Bc </sub>= 103</entry><entry>10</entry><entry>(5656)</entry><entry>(0.69)</entry></row><row><entry/><entry>λ/h ≈ 214</entry></row><row><entry/><entry>Q ≈ 3886</entry></row><row><entry/><entry namest="offset" nameend="4" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
Dielectric Disks
In some embodiments, one or more of the resonant objects are dielectric objects, such as disks. Consider a two dimensional dielectric disk object, as shown in FIG. 6, of radius r and relative permittivity c 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. HighQrad 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:
Qabs˜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="49pt" align="left"/><colspec colname="2" colwidth="35pt" 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/></row></tbody></tgroup><tgroup align="left" colsep="0" rowsep="0" cols="5"><colspec colname="1" colwidth="49pt" align="left"/><colspec colname="2" colwidth="35pt" align="char" char="."/><colspec colname="3" colwidth="49pt" align="center"/><colspec colname="4" colwidth="42pt" align="center"/><colspec colname="5" colwidth="42pt" align="center"/><tbody valign="top"><row><entry>Re{ε} = 147.7,</entry><entry>20.01</entry><entry>10103 (10075)</entry><entry>1988 (1992)</entry><entry>1661 (1663)</entry></row><row><entry>m = 2</entry><entry>(20.00)</entry></row><row><entry>Re{ε} = 65.6,</entry><entry>9.952</entry><entry>10098 (10087)</entry><entry>9078 (9168)</entry><entry>4780 (4802)</entry></row><row><entry>m = 3</entry><entry>(9.950)</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. 6. 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 Qrad≧2000 and Qabs˜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, Qrad=13962, and λ/r=17.
The required values of c, 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 c 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. 7 we place them at distance D between their centers. Numerically, the FDFD modesolver simulations give κ through the frequency splitting (=2κ) of the normal modes of the combined system, which are even and odd superpositions of the initial singledisk modes; analytically, using the expressions for the separationofvariables eigenfields E<sub>1,2</sub>(r) CMT gives κ through 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>where ∈<sub>j</sub>(r) and ∈(r) are the dielectric functions that describe only the disk j (minus the constant ∈<sub>0 </sub>background) and the whole space respectively. Then, for medium distances D/r=10−3 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, couplingtoloss ratios in the range κ/Γ˜150. Thus, for the analyzed embodiments, 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="49pt" align="left"/><colspec colname="2" colwidth="21pt" 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="49pt" align="left"/><colspec colname="2" colwidth="21pt" 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 λ/r ≈ 20</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>Q<sup>abs </sup>≈ 10093</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/><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 λ/r ≈ 10</entry><entry>5</entry><entry>9240</entry><entry>4824</entry><entry>2242 (2083)</entry><entry>2.2 (2.3)</entry></row><row><entry>Q<sup>abs </sup>≈ 10096</entry><entry>7</entry><entry>9187</entry><entry>4810</entry><entry>7485 (7417)</entry><entry>0.64 (0.65)</entry></row><row><entry namest="1" nameend="6" align="center" rowsep="1"/></row></tbody></tgroup></table></tables>
Note that even though particular embodiments 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 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 any one of these geometries, one can choose certain parameters to increase and/or optimize √{square root over (Q<sub>1</sub>Q<sub>2</sub>)}/Q<sub>κ</sub> or, if the Q's are limited by external factors, to increase and/or optimize for Q<sub>κ</sub>.
System Sensitivity to Extraneous Objects
In general, the overall performance of particular embodiment 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 object e the field amplitude a<sub>1</sub>(t) inside the resonant object 1 satisfies, to first order:
<maths id="MATHUS00040" num="00040"><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><msub><mi>κ</mi><mrow><mn>11</mn><mo></mo><mi>e</mi></mrow></msub><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mrow><mn>1</mn><mo></mo><mi>e</mi></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>11</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 κ<sub>11e </sub>is the frequency shift induced onto 1 due to the presence of e and Γ<sub>1e </sub>is the extrinsic due to e (absorption inside e, scattering from e etc.) loss rate. The firstorder PT model is valid only for small perturbations. Nevertheless, the parameters, κ<sub>11e</sub>, Γ<sub>1e </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 radiationΓ<sub>1e</sub>'s that are smaller than the initial radiationΓ<sub>1 </sub>(namely Γ<sub>1e </sub>is negative).
The frequency shift is a problem that can be “fixed” by applying to one or more resonant objects a feedback mechanism that corrects its frequency. For example, referring to FIG. 8a, in some embodiments each resonant object is provided with an oscillator at fixed frequency and a monitor which determines the frequency of the object. Both the oscillator and the monitor are coupled to a frequency adjuster which can adjust the frequency of the resonant object 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. The frequency adjuster determines the difference between the fixed frequency and the object frequency and acts to bring the object frequency into alignment with the fixed frequency. This technique assures that all resonant objects operate at the same fixed frequency, even in the presence of extraneous objects.
As another example, referring to FIG. 8b, in some embodiments, during energy transfer from a source object to a device object, the device object provides energy to a load, and an efficiency monitor measures the efficiency of the transfer. A frequency adjuster coupled to the load and the efficiency monitor acts to adjust the frequency of the object to maximize the transfer efficiency.
In various embodiments, other frequency adjusting schemes may be used which 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 may be transmitted to multiple devices, and each device then synched to that frequency.
Unlike the frequency shift, the extrinsic loss can be detrimental to the functionality of the energytransfer scheme, because it is difficult to remedy, so the total loss rate Γ<sub>1[e]</sub>=Γ<sub>1</sub>+Γ<sub>1e </sub>(and the corresponding figureofmerit κ<sub>[e]</sub>√{square root over (Γ<sub>1[e]</sub>Γ<sub>2[e]</sub>)}, where κ<sub>[e]</sub> the perturbed coupling rate) should be quantified.
CapacitivelyLoaded Conducting Loops or Coils
In embodiments using primarily magnetic resonances, the influence of extraneous objects on the resonances is nearly absent. The reason is that, in the quasistatic regime of operation (r<img id="CUSTOMCHARACTER00034" he="2.46mm" wi="1.44mm" file="US20110140544A120110616P00002.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>λ) that we are considering, the near field in the air region surrounding the resonator is predominantly magnetic (e.g. for coils with h<<2r most of the electric field is localized within the selfcapacitance of the coil or the externally loading capacitor), therefore extraneous nonconducting objects e that could interact with this field and act as a perturbation to the resonance are those having significant magnetic properties (magnetic permeability Re{μ}>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 on their surface.
As noted above, an extremely important implication of this fact relates to 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˜1 T 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 B10<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, it is reasonable to expect that our proposed energytransfer method should be safe for living organisms.
One can, for example, 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<<2) 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="MATHUS00041" num="00041"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><mfrac><mrow><msub><mi>u</mi><mi>e</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><msub><mi>u</mi><mi>m</mi></msub><mo></mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mrow><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><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><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><msup><mi>cos</mi><mn>2</mn></msup><mo></mo><mi>θ</mi></mrow><mo>+</mo><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></mrow></mfrac></mrow></mrow><mo>;</mo></mrow><mo></mo><mstyle><mtext></mtext></mstyle><mo></mo><mstyle><mspace width="1.1em" height="1.1ex"/></mstyle><mo></mo><mrow><mrow><mrow><mi>x</mi><mo>=</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><mi>π</mi><mo></mo><mfrac><msub><mi>D</mi><mi>p</mi></msub><mi>λ</mi></mfrac></mrow><mo>⇒</mo><mfrac><mrow><msub><mo>∯</mo><msub><mi>S</mi><mi>p</mi></msub></msub><mo></mo><mrow><mrow><msub><mi>u</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><msub><mo>∯</mo><msub><mi>S</mi><mi>p</mi></msub></msub><mo></mo><mrow><mrow><msub><mi>u</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></mrow><mo>;</mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo></mo><mrow><mi>x</mi><mo>=</mo><mrow><mn>2</mn><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></mrow></mtd><mtd><mrow><mo>(</mo><mn>12</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. (12) 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 may 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. (12) that for the loop of r=30 cm at a distance D<sub>p</sub>=10r=3 m 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=1 m 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, Γ<sub>1</sub>e<sup>abs</sup>=ω<sub>1</sub>/4·∫d<sup>3</sup>rIm{∈<sub>e</sub>(r)}E<sub>1</sub>(r)<sup>2</sup>/U 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×1.5 m 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 Q<sub>ch</sub><sup>abs</sup>˜10<sup>5 </sup>and for ˜10 cm away Q<sub>ch</sub><sup>abs</sup>˜5·10<sup>5</sup>. Thus, for ordinary distances (˜1 m) and placements (not immediately on top of the capacitor) or for most ordinary extraneous objects e of much smaller losstangent, we conclude that it is indeed fair to say that Q<sub>ce</sub><sup>abs</sup>→∞. The only perturbation that is expected to affect these resonances is a close proximity of large metallic structures.
Selfresonant coils are 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 are simple to make and can withstand much larger voltages than most lumped capacitors.
In general, different embodiments 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 u<sub>e</sub>/u<sub>m </sub>is reduced or minimized at most of the desired (according to the application) points in the surrounding space.
Dielectric Disks
In embodiments using resonances that are not primarily magnetic, the influence of extraneous objects may be of concern. For example, 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 All perturbations
<FORM>Γ<sub>1</sub>e<sup>rad</sup>=ω<sub>1</sub>∫d<sup>3</sup>rRe{∈<sub>e</sub>(r)}E<sub>1</sub>(r)<sup>2</sup>/U </FORM>
<FORM>and</FORM>
<FORM>Γ<sub>1</sub>e<sup>abs</sup>=ω<sub>1</sub>/4·∫d<sup>3</sup>rIm{∈<sub>e</sub>(r)}E<sub>1</sub>(r)<sup>2</sup>/U </FORM>
where U=½∫d<sup>3</sup>r∈(r)E<sub>1</sub>(r)<sup>2 </sup>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 rate from object 1 to another resonant object 2 is, as stated earlier,
<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>
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>>Γ<sub>1e</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 c 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. 9a, and a roughened surface of large extent but of small Re{∈}, Im{∈} (such as a wall w), as shown in FIG. 9b. For distances Dh/w/r=10<sup>−3 </sup>between the diskcenter and the “human”center or “wall”, the numerical FDFD simulation results presented in FIGS. 9a and 9b 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 FIG. 7 to FIG. 9c, the numerical FDFD simulations show that the system performance deteriorates from κ/Γ<sub>c</sub>˜150 to κ[hw]/Γ<sub>c[hw]</sub>˜0.510 i.e. only by acceptably small amounts.
Inductivelyloaded conducting rods may also be more sensitive than capacitivelyloaded coils, since they rely on the electric field to achieve the coupling.
System Efficiency
In general, another important factor for any energy transfer scheme is the transfer efficiency. Consider again the combined system of a resonant source s and device d in the presence of a set of extraneous objects e. The efficiency of this resonancebased energytransfer scheme may be determined, when energy is being drained from the device at rate Γ<sub>work </sub>for use into operational work. The coupledmodetheory equation for the device fieldamplitude is
<maths id="MATHUS00042" num="00042"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mfrac><mrow><mo></mo><msub><mi>a</mi><mi>d</mi></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><mi>ω</mi><mo></mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>Γ</mi><mrow><mi>d</mi><mo></mo><mrow><mo>[</mo><mi>e</mi><mo>]</mo></mrow></mrow></msub></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mo></mo><msub><mi>a</mi><mi>d</mi></msub></mrow><mo>+</mo><mrow><mi></mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>κ</mi><mrow><mo>[</mo><mi>e</mi><mo>]</mo></mrow></msub><mo></mo><msub><mi>a</mi><mi>s</mi></msub></mrow><mo></mo><mrow><msub><mi>Γ</mi><mi>work</mi></msub><mo></mo><msub><mi>a</mi><mi>d</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>13</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where Γ<sub>d[e]</sub>=Γ<sub>d[e]</sub><sup>rad</sup>+Γ<sub>d[e]</sub><sup>abs</sup>=Γ<sub>d[e]</sub><sup>rad</sup>+(Γ<sub>d</sub><sup>abs</sup>+Γ<sub>de</sub><sup>abs</sup>) is the net perturbeddevice loss rate, and similarly we define Γ<sub>s[e]</sub> for the perturbedsource. Different temporal schemes can be used to extract power from the device (e.g. steadystate continuouswave drainage, instantaneous drainage at periodic times and so on) and their efficiencies exhibit different dependence on the combined system parameters. For simplicity, we assume steady state, such that the field amplitude inside the source is maintained constant, namely a<sub>s</sub>(t)=A<sub>s</sub>e<sup>−iωt</sup>, so then the field amplitude inside the device is a<sub>d</sub>(t)=A<sub>d</sub>e<sup>−iωt </sup>with A<sub>d</sub>/A<sub>s</sub>=iκ<sub>[e]</sub>/(Γ<sub>d[e]</sub>+Γ<sub>work</sub>). The various timeaveraged powers of interest are then: the useful extracted power is P<sub>work</sub>=2Γ<sub>work</sub>A<sub>d</sub><sup>2</sup>, the radiated (including scattered) power is P<sub>rad</sub>=2Γ<sub>s[e]</sub><sup>rad</sup>A<sub>s</sub><sup>2</sup>+2Γ<sub>d[e]</sub><sup>rad</sup>A<sub>d</sub><sup>2</sup>, the power absorbed at the source/device is P<sub>s/d</sub>=2Γ<sub>s/d</sub><sup>abs</sup>A<sub>s/d</sub><sup>2</sup>, and at the extraneous objects P<sub>e</sub>=2Γ<sub>se</sub><sup>abs</sup>A<sub>s</sub><sup>2</sup>+2Γ<sub>de</sub><sup>abs</sup>A<sub>d</sub><sup>2</sup>. From energy conservation, the total timeaveraged power entering the system is P<sub>total</sub>=P<sub>work</sub>+P<sub>rad</sub>+P<sub>s</sub>+P<sub>d</sub>+P<sub>c</sub>. Note that the reactive powers, which are usually present in a system and circulate stored energy around it, cancel at resonance (which can be proven for example in electromagnetism from Poynting's Theorem) and do not influence the powerbalance calculations. The working efficiency is then:
<maths id="MATHUS00043" num="00043"><math overflow="scroll"><mtable><mtr><mtd><mrow><mrow><mrow><msub><mi>η</mi><mi>work</mi></msub><mo>≡</mo><mfrac><msub><mi>P</mi><mi>work</mi></msub><msub><mi>P</mi><mi>total</mi></msub></mfrac></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mrow><mfrac><msub><mi>Γ</mi><mrow><mi>d</mi><mo></mo><mrow><mo>[</mo><mi>e</mi><mo>]</mo></mrow></mrow></msub><msub><mi>Γ</mi><mi>work</mi></msub></mfrac><mo>·</mo><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><mrow><mfrac><mn>1</mn><msubsup><mi>fom</mi><mrow><mo>[</mo><mi>e</mi><mo>]</mo></mrow><mn>2</mn></msubsup></mfrac><mo></mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><msub><mi>Γ</mi><mi>work</mi></msub><msub><mi>Γ</mi><mrow><mi>d</mi><mo></mo><mrow><mo>[</mo><mi>c</mi><mo>]</mo></mrow></mrow></msub></mfrac></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mo>]</mo></mrow></mrow></mrow></mfrac></mrow><mo>,</mo></mrow></mtd><mtd><mrow><mo>(</mo><mn>14</mn><mo>)</mo></mrow></mtd></mtr></mtable></math></maths>
where fom<sub>[e]</sub>=κ<sub>[e]</sub>/√{square root over (Γ<sub>s[e]</sub>Γ<sub>d[e]</sub>)} is the distancedependent figureofmerit of the perturbed resonant energyexchange system. To derive Eq. (14), we have assumed that the rate Γ<sub>supply</sub>, at which the power supply is feeding energy to the resonant source, is such that there are zero reflections of the fed Γ<sub>supply</sub>=Γ<sub>s[e]</sub>+κ<sup>2</sup>/(Γ<sub>d[e]</sub>+Γ<sub>work</sub>), power P<sub>total </sub>back into the power supply.
Referring to FIG. 10, to rederive and express this formula (14) in terms of the parameters which are more directly accessible from particular resonant objects, e.g. 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<sub>s</sub>, 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 work (load) resistance R<sub>w </sub>to the device. The mutual inductance is denoted by M.
Then from the source circuit at resonance (ωL<sub>s</sub>=1/ωC<sub>s</sub>):
<maths id="MATHUS00044" num="00044"><math overflow="scroll"><mrow><mrow><msub><mi>V</mi><mi>g</mi></msub><mo>=</mo><mrow><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>jω</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>M</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msub><mi>I</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><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>jω</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><mi>M</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo></mo><msubsup><mi>I</mi><mi>d</mi><mo>*</mo></msubsup><mo></mo><msub><mi>I</mi><mi>s</mi></msub></mrow></mrow></mrow></mrow><mo>,</mo></mrow></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>w</sub>)−jωMI<sub>s</sub><img id="CUSTOMCHARACTER00035" he="3.13mm" wi="2.79mm" file="US20110140544A120110616P00007.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>jωMI<sub>s</sub>=I<sub>d</sub>(R<sub>d</sub>+R<sub>w</sub>)</FORM>
So by substituting the second to the first:
<maths id="MATHUS00045" num="00045"><math overflow="scroll"><mrow><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><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><mrow><mo>(</mo><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>w</mi></msub></mrow><mo>)</mo></mrow><mo>.</mo></mrow></mrow></mrow></mrow></math></maths>
Now we take the real part (timeaveraged powers) to find the efficiency:
<maths id="MATHUS00046" num="00046"><math overflow="scroll"><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><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>w</mi></msub></mrow><mo>⇒</mo><msub><mi>η</mi><mi>work</mi></msub></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>≡</mo><mi/><mo></mo><mfrac><msub><mi>P</mi><mi>w</mi></msub><msub><mi>P</mi><mi>tot</mi></msub></mfrac></mrow><mo>=</mo><mrow><mfrac><msub><mi>R</mi><mi>w</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><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>w</mi></msub></mrow></mfrac><mo>.</mo></mrow></mrow></mtd></mtr></mtable></math></maths>
Namely<maths id="MATHUS00047" num="00047"><math overflow="scroll"><mrow><mrow><msub><mi>η</mi><mi>work</mi></msub><mo>=</mo><mfrac><msub><mi>R</mi><mi>w</mi></msub><mrow><mrow><mfrac><msup><mrow><mo>(</mo><mrow><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>w</mi></msub></mrow><mo>)</mo></mrow><mn>2</mn></msup><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></mfrac><mo>·</mo><msub><mi>R</mi><mi>s</mi></msub></mrow><mo>+</mo><msub><mi>R</mi><mi>d</mi></msub><mo>+</mo><msub><mi>R</mi><mi>w</mi></msub></mrow></mfrac></mrow><mo>,</mo></mrow></math></maths>
which with Γ<sub>work</sub>=R<sub>w</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>)}, becomes the general Eq. (14).
End of ExampleFrom Eq. (14) one can find that the efficiency is optimized in terms of the chosen workdrainage rate, when this is chosen to be Γ<sub>work</sub>/Γ<sub>d[e]</sub>=Γ<sub>supply</sub>/Γ<sub>s[e]</sub>=√{square root over (1+fom<sub>[c]</sub><sup>2</sup>)}>1. Then, η<sub>work </sub>is a function of the fom<sub>[e]</sub> parameter only as shown in FIG. 11 with a solid black line. One can see that the efficiency of the system is η>17% for fom<sub>[e]</sub>>1, large enough for practical applications. Thus, the efficiency can be further increased towards 100% by optimizing fom<sub>[e]</sub> as described above. The ratio of conversion into radiation loss depends also on the other system parameters, and is plotted in FIG. 5 for the conducting loops with values for their parameters within the ranges determined earlier.
For example, consider the capacitivelyloaded coil embodiments described in Table 4, with coupling distance D/r=7, a “human” extraneous object at distance D<sub>h </sub>from the source, and that P<sub>work</sub>=10 W must be delivered to the load. Then, we have (based on FIG. 11) Q<sub>s[h]</sub><sup>rad</sup>=Q<sub>d[h]</sub><sup>rad</sup>˜10<sup>4</sup>, Q<sub>s</sub><sup>abs</sup>=Q<sub>d</sub><sup>abs</sup>˜10<sup>3</sup>, Q<sub>κ</sub>˜500, and Q<sub>dh</sub><sup>abs</sup>→∞, Q<sub>sh</sub><sup>abs</sup>˜10<sup>5 </sup>at D<sub>h</sub>˜3 cm and Q<sub>sh</sub><sup>abs</sup>˜5·10<sup>5 </sup>at D<sub>h</sub>˜10 cm. Therefore fom<sub>[h]</sub>˜2, so we find η≈38%, P<sub>rad</sub>≈1.5 W, P<sub>s</sub>≈11 W, P<sub>d</sub>≈4 W, and most importantly η<sub>h</sub>≈0.4%, P<sub>h</sub>=0.1 W at D<sub>h</sub>˜3 cm and η<sub>h</sub>≈0.1%, P<sub>h</sub>=0.02 W at D<sub>h</sub>˜10 cm.
Overall System Performance
In many cases, the dimensions of the resonant objects will be set by the particular application at hand. For example, when this application is powering a laptop or a cellphone, the device resonant object cannot have dimensions larger that those of the laptop or cellphone respectively. In particular, 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 workextraction rate (load resistance) Γ<sub>work </sub>and the powersupply feeding rate Γ<sub>supply</sub>.
In general, in various embodiments, the primary dependent variable that one wants to increase or optimize is the overall efficiency η. However, other important variables need to be taken into consideration upon system design. For example, in embodiments featuring capacitivelyloaded coils, the design may be constrained by, for example, the currents flowing inside the wires I<sub>s,d </sub>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 Q<sub>tot</sub>=ωL<sub>d</sub>/(R<sub>d</sub>+R<sub>w</sub>) of the device is a quantity 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>rad,s,d </sub>should be minimized for safety concerns, 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 workdrainage rate for some particular value of fom<sub>[d]</sub> through Γ<sub>work</sub>/Γ<sub>d[c]</sub>=√{square root over (1+wp·fom<sub>[e]</sub><sup>2</sup>)}. Then, in some embodiments, values which impact the choice of this rate are: Γ<sub>work</sub>/Γ<sub>d[e]</sub>=1<img id="CUSTOMCHARACTER00036" he="2.46mm" wi="3.13mm" file="US20110140544A120110616P00004.TIF" alt="customcharacter" 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>), Γ<sub>work</sub>/Γ<sub>d[e]</sub>=√{square root over (1+fom<sub>[c]</sub><sup>2</sup>)}>1<img id="CUSTOMCHARACTER00037" he="2.46mm" wi="3.13mm" file="US20110140544A120110616P00004.TIF" alt="customcharacter" imgcontent="character" imgformat="tif"/>wp=1 to increase the efficiency, as seen earlier, or Γ<sub>work</sub>/Γ<sub>d[e]</sub>>>1<img id="CUSTOMCHARACTER00038" he="2.46mm" wi="3.13mm" file="US20110140544A120110616P00004.TIF" alt="customcharacter" 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>tot</sub>=ωL<sub>d</sub>/(R<sub>d</sub>+R<sub>w</sub>)=ω/[2(Γ<sub>d</sub>+Γ<sub>work</sub>)]. Similar is the impact of the choice of the power supply feeding rate Γ<sub>supply</sub>, with the roles of the source and the device reversed.
Increasing N<sub>s </sub>and N<sub>d </sub>increases κ/√{square root over (Γ<sub>s</sub>Γ<sub>d</sub>)} and thus efficiency significantly, as seen before, and also decreases the currents I<sub>s </sub>and I<sub>d</sub>, because the inductance of the loops increases, and thus the energy
<maths id="MATHUS00048" num="00048"><math overflow="scroll"><mrow><msub><mi>U</mi><mrow><mi>s</mi><mo>,</mo><mi>d</mi></mrow></msub><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo></mo><msub><mi>L</mi><mrow><mi>s</mi><mo>,</mo><mi>d</mi></mrow></msub><mo></mo><msup><mrow><mo></mo><msub><mi>I</mi><mrow><mi>s</mi><mo>,</mo><mi>d</mi></mrow></msub><mo></mo></mrow><mn>2</mn></msup></mrow></mrow></math></maths>
required for given output power P<sub>work </sub>can be achieved with smaller currents. However, increasing N<sub>d </sub>increases Q<sub>tot</sub>, P<sub>rad,d </sub>and the voltage across the device capacitance V<sub>d</sub>, which unfortunately ends up being, in typical embodiments one of the greatest limiting factors of the system. To explain this, note that it is the electric field that really induces breakdown of the capacitor material (e.g. 3 kV/mm for air) and not the voltage, and that for the desired (close to the optimal) operational frequency, the increased inductance L<sub>d </sub>implies reduced required capacitance C<sub>d</sub>, which could be achieved in principle, for a capacitivelyloaded device coil by increasing the spacing of the device capacitor plates d<sub>d </sub>and for a selfresonant coil by increasing through h<sub>d </sub>the spacing of adjacent turns, resulting in an electric field (≈V<sub>d</sub>/d<sub>d </sub>for the former case) that actually decreases with N<sub>d</sub>; however, one cannot in reality increase d<sub>d </sub>or h<sub>d </sub>too much, because then the undesired capacitance fringing electric fields would become very large and/or the size of the coil might become too large; and, in any case, for certain applications extremely high voltages are not desired. A similar increasing behavior is observed for the source P<sub>rad,s </sub>and V<sub>s </sub>upon increasing N<sub>s</sub>. As a conclusion, the number of turns N<sub>s </sub>and N<sub>d </sub>have to be chosen the largest possible (for efficiency) that allow for reasonable voltages, fringing electric fields and physical sizes.
With respect to frequency, again, there is an optimal one for efficiency, and Q<sub>tot </sub>is approximately maximum, close to that optimal frequency. For lower frequencies the currents get worse (larger) but the voltages and radiated powers get better (smaller). Usually, one should pick either the optimal frequency or somewhat lower.
One way to decide on an operating regime for the system is based on a graphical method. In FIG. 12, for two loops of r<sub>s</sub>=25 cm, r<sub>d</sub>=15 cm, h<sub>s</sub>=h<sub>d</sub>=0, a<sub>s</sub>=a<sub>d</sub>=3 mm and distance D=2 m between them, we plot all the above dependent variables (currents, voltages and radiated powers normalized to 1 Watt of output power) in terms of frequency and N<sub>d</sub>, given some choice for wp and N<sub>s</sub>. The Figure depicts all of the dependencies explained above. We can also 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. The results are shown in FIG. 13 for the same loop dimensions and distance. For example, a reasonable choice of parameters for the system of two loops with the dimensions given above are: N<sub>s</sub>=2, N<sub>d</sub>=6, f=10 MHz and wp=10, which gives the following performance characteristics: η<sub>work</sub>=20.6%, Q<sub>tot</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>rad,s</sub>=0.15 W, P<sub>rad,d</sub>=0.006 W. Note that the results in FIGS. 12 and 13, and the just above 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>, still they give a good estimate of the scalings and the orders of magnitude.
Finally, one could additionally optimize for the source dimensions, since usually only the device dimensions are limited, as discussed earlier. 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 too for all the dependent variables of the problem (we saw how to do this only for efficiency earlier). Such an optimization would lead to improved results.
EXPERIMENTAL RESULTSAn experimental realization of an embodiment of the above described scheme for wireless energy transfer consists of two selfresonant coils of the type described above, 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. 14. Referring to FIG. 14, 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, 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 about 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 coupledmode theory, the splitting in the transmission spectrum should be Δω=2√{square root over (κ<sup>2</sup>−Γ<sup>2</sup>)}. The comparison between experimental and theoretical results as a function of distance when the two the coils are aligned coaxially is shown in FIG. 15.
FIG. 16 shows a comparison of experimental and theoretical values for the parameter 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 κ/Γ due to the ˜5% uncertainty in Q.
As noted above, the maximum theoretical efficiency depends only on the parameter κ/√{square root over (Γ<sub>1</sub>Γ<sub>2</sub>)}=κ/Γ, plotted as a function of distance in FIG. 17. The coupling to loss ratio κ/Γ is greater than 1 even for D=2.4 m (eight times the radius of the coils), thus the system is in the stronglycoupled regime throughout the entire range of distances probed.
The power supply 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. 14). 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 Γ<sub>work</sub>/Γ was adjusted so that it matched its optimal value, given theoretically by √{square root over (1+κ<sup>2</sup>/(Γ<sub>1</sub>Γ<sub>2</sub>))}. Because of its inductive nature, the loop connected to the lightbulb added a small reactive component to Γ<sub>work </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</sub><sup>2</sup>, and the efficiency was directly obtained from η=P<sub>work</sub>/(P<sub>1</sub>+P<sub>2</sub>+P<sub>work</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. 17, along with the theoretical prediction for maximum efficiency, given by Eq. (14).
Using this embodiment, we were able to transfer significant amounts of power using this setup, 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 transfer 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 transferring 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 transfer at that distance and the low efficiency of the driving circuit.
From the theoretical treatment above, we see that in typical embodiments it is important that the coils be on resonance for the power transfer 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 transfer 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.
We believe that this method of power transfer should be safe for humans. When transferring 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 transfer over medium range distances, and found that the experimental results match theory well in multiple independent and mutually consistent tests.
We believe that the efficiency of the scheme and the distances covered could be appreciably improved by silverplating the coils, which should 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.
APPLICATIONSIn conclusion, we have described several embodiments of a resonancebased scheme for wireless nonradiative energy transfer. Although our consideration has been for a static geometry (namely κ and Γ<sub>e </sub>were independent of time), all the results can be applied directly for the dynamic geometries of mobile objects, since the energytransfer time κ<sup>−1 </sup>(˜1 μs−1 ms for microwave applications) is much shorter than any timescale associated with motions of macroscopic objects. Analyses of very simple implementation geometries provide encouraging performance characteristics and further improvement is expected with serious design optimization. Thus the proposed mechanism is promising for many modem applications.
For example, in the macroscopic world, this scheme could potentially be used to deliver power to for example, robots and/or computers in a factory room, or electric buses on a highway. In some embodiments sourceobject could be an elongated “pipe” running above the highway, or along the ceiling.
Some embodiments of the wireless transfer scheme can provide energy to power or charge devices that are difficult or impossible to reach using wires or other techniques. For example some embodiments may provide power to implanted medical devices (e.g. artificial hearts, pacemakers, medicine delivery pumps, etc.) or buried underground sensors.
In the microscopic world, where much smaller wavelengths would be used and smaller powers are needed, one could use it to implement optical interconnects for CMOS electronics, or to transfer energy to autonomous nanoobjects (e.g. MEMS or nanorobots) without worrying much about the relative alignment between the sources and the devices. Furthermore, the range of applicability could be extended to acoustic systems, where the source and device are connected via a common condensedmatter object.
In some embodiments, the techniques described above can provide nonradiative wireless transfer of information using the localized near fields of resonant object. Such schemes provide increased security because no information is radiated into the farfield, and are well suited for midrange communication of highly sensitive information.
A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention.