U.S. patent application number 16/137795 was filed with the patent office on 2019-04-25 for wireless energy transfer.
The applicant listed for this patent is Massachusetts Institute of Technology. Invention is credited to Peter H. Fisher, John D. Joannopoulos, Aristeidis Karalis, Andre B. Kurs, Robert Moffatt, Marin Soljacic.
Application Number | 20190123586 16/137795 |
Document ID | / |
Family ID | 46330224 |
Filed Date | 2019-04-25 |
View All Diagrams
United States Patent
Application |
20190123586 |
Kind Code |
A1 |
Karalis; Aristeidis ; et
al. |
April 25, 2019 |
WIRELESS ENERGY TRANSFER
Abstract
Disclosed is an apparatus for use in wireless energy transfer,
which includes a first resonator structure configured to transfer
energy non-radiatively with a second resonator structure over a
distance greater than a characteristic size of the second resonator
structure. The non-radiative energy transfer is mediated by a
coupling of a resonant field evanescent tail of the first resonator
structure and a resonant field evanescent tail of the second
resonator structure.
Inventors: |
Karalis; Aristeidis;
(Boston, MA) ; Kurs; Andre B.; (Chestnut Hill,
MA) ; Moffatt; Robert; (Reston, VA) ;
Joannopoulos; John D.; (Belmont, MA) ; Fisher; Peter
H.; (Cambridge, MA) ; Soljacic; Marin;
(Belmont, MA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Massachusetts Institute of Technology |
Cambridge |
MA |
US |
|
|
Family ID: |
46330224 |
Appl. No.: |
16/137795 |
Filed: |
September 21, 2018 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
15186969 |
Jun 20, 2016 |
10097044 |
|
|
16137795 |
|
|
|
|
13789860 |
Mar 8, 2013 |
9509147 |
|
|
15186969 |
|
|
|
|
13477459 |
May 22, 2012 |
9444265 |
|
|
13789860 |
|
|
|
|
13036177 |
Feb 28, 2011 |
|
|
|
13477459 |
|
|
|
|
12437641 |
May 8, 2009 |
8097983 |
|
|
13036177 |
|
|
|
|
12055963 |
Mar 26, 2008 |
7825543 |
|
|
12437641 |
|
|
|
|
11481077 |
Jul 5, 2006 |
7741734 |
|
|
12055963 |
|
|
|
|
PCT/US2007/070892 |
Jun 11, 2007 |
|
|
|
11481077 |
|
|
|
|
60698442 |
Jul 12, 2005 |
|
|
|
60908383 |
Mar 27, 2007 |
|
|
|
60908666 |
Mar 28, 2007 |
|
|
|
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
H02J 5/005 20130101;
H02J 50/90 20160201; Y02T 90/127 20130101; Y10T 29/4902 20150115;
H02J 7/025 20130101; B60L 53/126 20190201; Y02T 10/70 20130101;
Y02T 90/122 20130101; Y02T 90/14 20130101; Y02T 10/725 20130101;
Y02T 10/7072 20130101; Y02T 90/12 20130101; Y02T 10/7005 20130101;
H01Q 9/04 20130101; H04B 5/0037 20130101; B60L 2210/20 20130101;
H01Q 7/00 20130101; Y02T 10/72 20130101; H02J 50/12 20160201; H02J
50/80 20160201 |
International
Class: |
H02J 50/12 20060101
H02J050/12; H02J 50/90 20060101 H02J050/90; H02J 7/02 20060101
H02J007/02; H01Q 7/00 20060101 H01Q007/00; H01Q 9/04 20060101
H01Q009/04; H04B 5/00 20060101 H04B005/00; H02J 5/00 20060101
H02J005/00; H02J 50/80 20060101 H02J050/80 |
Goverment Interests
STATEMENT AS TO FEDERALLY FUNDED RESEARCH
[0002] This invention was made with government support awarded by
the National Science Foundation under Grant No. DMR 02-13282. The
government has certain rights in this invention.
Claims
1. A power supply system comprising: a power supply loop that is
supplied with power from a power supply; a power supply-side
resonance coil that is supplied with power from the power supply
loop in a non-contact manner; a power receiving-side resonance coil
that is supplied with power from the power supply-side resonance
coil in a non-contact manner; a power receiving loop that is
supplied with power from the power receiving-side resonance coil in
a non-contact manner; wherein at least one of a distance between
the power supply loop and the power supply-side resonance coil and
a distance between the power receiving loop and the power
receiving-side resonance coil is adjustable on the basis of a
distance between the power supply-side resonance coil and the power
receiving-side resonance coil.
Description
[0001] This application is a continuation and claims the benefit of
priority under 35 USC .sctn. 120 to U.S. application Ser. No.
15/186,969, filed Jun. 20, 2016, which is a continuation of U.S.
application Ser. No. 13/789,860, filed Mar. 8, 2013, now U.S. Pat.
No. 9,509,147, which is a continuation of U.S. application Ser. No.
13/477,459, filed May 22, 2012, now U.S. Pat. No. 9,444,265, which
is a continuation of U.S. application Ser. No. 13/036,177, filed
Feb. 28, 2011, which is a continuation of U.S. application Ser. No.
12/437,641, filed May 8, 2009, now U.S. Pat. No. 8,097,983, which
is a continuation of U.S. application Ser. No. 12/055,963 filed
Mar. 26, 2008, now U.S. Pat. No. 7,825,543, which: 1) is a
continuation-in-part of U.S. Utility patent application Ser. No.
11/481,077, filed Jul. 5, 2006, now U.S. Pat. No. 7,741,734, which
claims priority to U.S. Provisional Application Ser. No.
60/698,442, filed Jul. 12, 2005; (2) pursuant to U.S.C. .sctn.
119(e), U.S. application Ser. No. 12/055,963 claims priority to
U.S. Provisional Application Ser. No. 60/908,383, filed Mar. 27,
2007, and U.S. Provisional Application Ser. No. 60/908,666, filed
Mar. 28, 2007; and (3) pursuant to U.S.C. .sctn. 120, and U.S.C.
.sctn. 363, U.S. application Ser. No. 12/055,963 is also a
continuation-in-part of International Application No.
PCT/US2007/070892, filed Jun. 11, 2007. The contents of the prior
applications are incorporated herein by reference in their
entirety.
BACKGROUND
[0003] The 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.
[0004] Radiative modes of omni-directional 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
highly-directional antennas, can be efficiently used for energy
transfer, even for long distances (transfer distance
L.sub.TRANS>>L.sub.DEV, where L.sub.DEV is the characteristic
size of the device and/or the source), but require existence of an
uninterruptible line-of-sight and a complicated tracking system in
the case of mobile objects. Some transfer schemes rely on
induction, but are typically restricted to very close-range
(L.sub.TRANS<<L.sub.DEV) or low power (.about.mW) energy
transfers.
[0005] The rapid development of autonomous electronics of recent
years (e.g. laptops, cell-phones, house-hold robots, that all
typically rely on chemical energy storage) has led to an increased
need for wireless energy transfer.
SUMMARY
[0006] The inventors have realized that resonant objects with
coupled resonant modes having localized evanescent field patterns
may be used for non-radiative wireless energy transfer. Resonant
objects tend to couple, while interacting weakly with other
off-resonant 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
energy-transfer rate can be larger than the energy-loss rate.
Accordingly, efficient wireless energy-exchange can be achieved
between the resonant objects, while suffering only modest transfer
and dissipation of energy into other off-resonant objects. The
nearly-omnidirectional but stationary (non-lossy) 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 electric-engine buses and/or hybrid cars
and medical implantable devices.
[0007] In some embodiments, resonant modes are so-called 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 non-magnetic,
their interaction with magnetic fields is minimal. This is
important both for safety and also to reduce interaction with the
extraneous environmental objects.
[0008] 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 of the
second resonator structure. In some embodiments, D is also greater
than one or more of: a characteristic size L.sub.1 of the first
resonator structure, a characteristic thickness T.sub.1 of the
first resonator structure, and a characteristic width W.sub.1 of
the first resonator structure. The energy transfer is mediated by
evanescent-tail 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.
[0009] 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.
[0010] In some embodiments, the first resonator structure has a
resonant angular frequency co, a Q-factor Q.sub.1, and a resonance
width .GAMMA..sub.1, the second resonator structure has a resonant
angular frequency co, a Q-factor Q.sub.2, and a resonance width
.GAMMA..sub.2, and the energy transfer has a rate .kappa.. In some
embodiments, the absolute value of the difference of the angular
frequencies .omega..sub.1 and .omega..sub.2 is smaller than the
broader of the resonant widths .GAMMA..sub.1 and .GAMMA..sub.2.
[0011] In some embodiments Q.sub.1>100 and Q.sub.2>100,
Q.sub.1>300 and Q.sub.2>300, Q.sub.1>500 and
Q.sub.2>500, or Q.sub.1>1000 and Q.sub.2>1000. In some
embodiments, Q.sub.1>100 or Q.sub.2>100, Q.sub.1>300 or
Q.sub.2>300, Q.sub.1>500 or Q.sub.2>500, or
Q.sub.1>1000 or Q.sub.2>1000.
[0012] In some embodiments, the coupling to loss ratio
.kappa. .GAMMA. 1 .GAMMA. 2 > 0.5 , .kappa. .GAMMA. 1 .GAMMA. 2
> 1 , .kappa. .GAMMA. 1 .GAMMA. 2 > 2 , or .kappa. .GAMMA. 1
.GAMMA. 2 > 5. ##EQU00001##
In some such embodiments, D/L.sub.2 may be as large as 2, as large
as 3, as large as 5, as large as 7, or as large as 10.
[0013] In some embodiments, Q.sub.1>1000 and Q.sub.2>1000,
and the coupling to loss ratio
.kappa. .GAMMA. 1 .GAMMA. 2 > 10 , .kappa. .GAMMA. 1 .GAMMA. 2
> 25 , or .kappa. .GAMMA. 1 .GAMMA. 2 > 40. ##EQU00002##
In some such embodiments, D/L.sub.2 may be as large as 2, as large
as 3, as large as 5, as large as 7, as large as 10.
[0014] In some embodiments, Q.sub..kappa.=.omega./2.kappa. 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 is as large as
2, as large as 3, as large as 5, as large as 7, or as large as
10.
[0015] In some embodiments, the quantity
.kappa./.GAMMA..sub.1.GAMMA..sub.2 is maximized at an angular
frequency {tilde over (.omega.)} with a frequency width {tilde over
(.GAMMA.)} around the maximum, and the absolute value of the
difference of the angular frequencies .omega..sub.1 and {tilde over
(.omega.)} is smaller than the width {tilde over (.GAMMA.)}, and
the absolute value of the difference of the angular frequencies
{tilde over (.omega.)}.sub.2 and {tilde over (.omega.)} is smaller
than the width {tilde over (.GAMMA.)}.
[0016] In some embodiments, the energy transfer operates with an
efficiency .eta..sub.work greater than about 1%, greater than about
10%, greater than about 30%, greater than about 50%, or greater
than about 80%.
[0017] In some embodiments, the energy transfer operates with a
radiation loss .eta..sub.rad less that about 10%. In some such
embodiments the coupling to loss ratio
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 0.1 . ##EQU00003##
[0018] In some embodiments, the energy transfer operates with a
radiation loss .eta..sub.rad less that about 1%. In some such
embodiments, the coupling to loss ratio
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 1. ##EQU00004##
[0019] 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 .eta..sub.h of
less than about 1%. In some such embodiments the coupling to loss
ratio
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 1. ##EQU00005##
[0020] 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 .eta..sub.h of
less than about 0.2%. In some such embodiments the coupling to loss
ratio
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 1. ##EQU00006##
[0021] In some embodiments, during operation, a device coupled to
the first or second resonator structure with a coupling rate
.GAMMA..sub.work receives a usable power P.sub.work from the
resonator structure.
[0022] In some embodiments, P.sub.work is greater than about 0.01
Watt, greater than about 0.1 Watt, greater than about 1 Watt, or
greater than about 10 Watt.
[0023] In some embodiments, if the device is coupled to the first
resonator, then
1/2[(.GAMMA..sub.work/.GAMMA..sub.1).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.2, or
1/4[(.GAMMA..sub.work/.GAMMA..sub.1).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.4, or
1/8[(.GAMMA..sub.work/.GAMMA..sub.1).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.8, and, if
the device is coupled to the second resonator, then
1/2[(.GAMMA..sub.work/.GAMMA..sub.2).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.2, or
1/4[(.GAMMA..sub.work/.GAMMA..sub.2).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.4, or
1/8[(.GAMMA..sub.work/.GAMMA..sub.1).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.8.
[0024] 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 nano-robot). In some embodiments,
the device includes a mobile electronic device (e.g. a telephone,
or a cell-phone, 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).
[0025] In some embodiments, the apparatus further includes the
device.
[0026] In some embodiments, during operation, a power supply
coupled to the first or second resonator structure with a coupling
rate .GAMMA..sub.supply drives the resonator structure at a
frequency f and supplies power P.sub.total. In some embodiments,
the absolute value of the difference of the angular frequencies
.omega.=2.pi.f and .omega..sub.1 is smaller than the resonant width
.GAMMA..sub.1, and the absolute value of the difference of the
angular frequencies .omega.=2.pi.f and .omega..sub.2 is smaller
than the resonant width .GAMMA..sub.2. In some embodiments, f is
about the optimum efficiency frequency.
[0027] In some embodiments, if the power supply is coupled to the
first resonator, then
1/2[(.GAMMA..sub.supply/.GAMMA..sub.1).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.2, or
1/4[(.GAMMA..sub.supply/.GAMMA..sub.1).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.4, or
1/8[(.GAMMA..sub.supply/.GAMMA..sub.1).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.8, and, if
the power supply is coupled to the second resonator, then
1/2[(.GAMMA..sub.supply/.GAMMA..sub.2).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.2, or
1/4[(.GAMMA..sub.supply/.GAMMA..sub.2).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.4, or
1/8[(.GAMMA..sub.supply/.GAMMA..sub.2).sup.2-1]/(.kappa./ {square
root over (.GAMMA..sub.1.GAMMA..sub.2)}).sub.2.ltoreq.8.
[0028] In some embodiments, the apparatus further includes the
power source.
[0029] 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.
[0030] 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 from the
closest resonant object is less than 0.01, or less than 0.1. In
some embodiments, L.sub.R is the characteristic size of the closest
resonant object and D.sub.p/L.sub.R is less than 1.5, 3, 5, 7, or
10.
[0031] 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.
[0032] 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>300 and Q.sub.2>300.
[0033] 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.sub.2-h.sup.2)}/2.pi.r.
[0034] 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
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 40 , .kappa. .GAMMA. 1 .GAMMA.
2 .gtoreq. 15 , or .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 5 , or
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 1. ##EQU00007##
In some such embodiments D/L.sub.R is as large as about 2, 3, 5, or
8.
[0035] 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
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 70 , .kappa. .GAMMA. 1 .GAMMA.
2 .gtoreq. 19 , or .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 8 , or
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 3. ##EQU00008##
In some such embodiments D/L.sub.R is as large as about 3, 5, 7, or
10.
[0036] 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
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 59 , .kappa. .GAMMA. 1 .GAMMA.
2 .gtoreq. 15 , or .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 6 , or
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 2. ##EQU00009##
In some such embodiments D/L.sub.R is as large as about 3, 5, 7, or
10.
[0037] 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>300 and
Q.sub.2>300.
[0038] In some embodiments, the characteristic size L.sub.R 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
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 14.9 , .kappa. .GAMMA. 1
.GAMMA. 2 .gtoreq. 3.2 , .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 1.2 ,
or .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 0.4 . ##EQU00010##
In some such embodiments, D/L.sub.R is as large as about 3, about
5, about 7, or about 10.
[0039] In some embodiments, the characteristic size of the
resonator structure receiving energy from the other resonator
structure L.sub.R 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
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 15.9 , .kappa. .GAMMA. 1
.GAMMA. 2 .gtoreq. 4.3 , .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 1.8 ,
or .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 0.7 . ##EQU00011##
In some such embodiments, D/L.sub.R is as large as about 3, about
5, about 7, or about 10.
[0040] In some embodiments, the characteristic size L.sub.R 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.
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 67.4 , .kappa. .GAMMA. 1
.GAMMA. 2 .gtoreq. 17.8 , .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 7.1
, or ##EQU00012## .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 2.7 .
##EQU00012.2##
In some such embodiments, D/L.sub.R is as large as about 3, about
5, about 7, or about 10.
[0041] In some embodiments, the characteristic size of the
resonator structure receiving energy from the other resonator
structure L.sub.R 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
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 6.3 , .kappa. .GAMMA. 1
.GAMMA. 2 .gtoreq. 1.3 , .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 0.5 .
, or .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 0.2 . ##EQU00013##
In some such embodiments, D/L.sub.R is as large as about 3, about
5, about 7, or about 10.
[0042] In some embodiments, the characteristic size L.sub.R 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 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 5 MHz. In some such
embodiments, the coupling to loss ratio
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 6.8 , .kappa. .GAMMA. 1
.GAMMA. 2 .gtoreq. 1.4 , .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 0.5 ,
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 0.2 . ##EQU00014##
In some such embodiments, D/L.sub.R is as large as about 3, about
5, about 7, or about 10.
[0043] In some embodiments, during operation, one of the resonator
structures receives a usable power P from the other resonator
structure, an electrical current I.sub.s flows in the resonator
structure which is transferring energy to the other resonant
structure, and the ratio
I s P w ##EQU00015##
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 from the other resonator structure, a voltage
difference V.sub.s appears across the capacitive element of the
first resonator structure, and the ratio
V s P w ##EQU00016##
is less than about 2000 Volts/ {square root over (Watts)} or less
than about 4000 Volts/ {square root over (Watts)}.
[0044] 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.1>300 and Q.sub.2>300.
[0045] In some embodiments, the characteristic size of the
resonator structure receiving energy from the other resonator
structure L.sub.R 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
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 32 , .kappa. .GAMMA. 1 .GAMMA.
2 .gtoreq. 5.8 , .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 2 , or
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 0.6 . ##EQU00017##
In some such embodiments, D/L.sub.R is as large as about 3, about
5, about 7, or about 10.
[0046] In some embodiments, the characteristic size L.sub.R 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
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 105 , .kappa. .GAMMA. 1
.GAMMA. 2 .gtoreq. 19 , .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 6.6 ,
or .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 2.2 . ##EQU00018##
In some such embodiments, D/L.sub.R is as large as about 3, about
5, about 7, or about 10.
[0047] 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>300 and
Q.sub.2>300.
[0048] In some embodiments, the characteristic size of the
resonator structure receiving energy from the other resonator
structure is L.sub.R and the real part of the permittivity of said
resonator structure .epsilon. is less than about 150. In some such
embodiments, the coupling to loss ratio
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 42.4 , .kappa. .GAMMA. 1
.GAMMA. 2 .gtoreq. 6.5 , .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 2.3 ,
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 0.5 . ##EQU00019##
In some such embodiments, D/L.sub.R is as large as about 3, about
5, about 7, or about 10.
[0049] In some embodiments, the characteristic size of the
resonator structure receiving energy from the other resonator
structure is L.sub.R and the real part of the permittivity of said
resonator structure .epsilon. is less than about 65. In some such
embodiments, the coupling to loss ratio
.kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq. 30.9 , .kappa. .GAMMA. 1
.GAMMA. 2 .gtoreq. 2.3 , or .kappa. .GAMMA. 1 .GAMMA. 2 .gtoreq.
0.5 . ##EQU00020##
In some such embodiments, D/L.sub.R is as large as about 3, about
5, about 7.
[0050] 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.
[0051] In some embodiments, at least one of the resonators has a
quality factor greater than about 5000, or greater than about
10000.
[0052] 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
evanescent-tail 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.
[0053] In some embodiments, the third resonator structure is
configured to transfer energy to one or more of the first and
second resonator structures.
[0054] In some embodiments, the third resonator structure is
configured to receive energy from one or more of the first and
second resonator structures.
[0055] In some embodiments, the third 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.
[0056] 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.
[0057] 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 of the
second resonator structure. In some embodiments, D is also greater
than one or more of: a characteristic size L.sub.1 of the first
resonator structure, a characteristic thickness T.sub.1 of the
first resonator structure, and a characteristic width W.sub.1 of
the first resonator structure. The energy transfer is mediated by
evanescent-tail coupling of a resonant field of the first resonator
structure and a resonant field of the second resonator
structure.
[0058] 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.
[0059] In some embodiments, the first resonator structure has a
resonant angular frequency .omega..sub.1, a Q-factor Q.sub.1, and a
resonance width .GAMMA..sub.1, the second resonator structure has a
resonant angular frequency .omega..sub.2, a Q-factor Q.sub.2, and a
resonance width .GAMMA..sub.2, and the energy transfer has a rate
.kappa.. In some embodiments, the absolute value of the difference
of the angular frequencies .omega..sub.1 and .omega..sub.2 is
smaller than the broader of the resonant widths .GAMMA..sub.1 and
.GAMMA..sub.2.
[0060] In some embodiments, the coupling to loss ratio
.kappa. .GAMMA. 1 .GAMMA. 2 > 0.5 , .kappa. .GAMMA. 1 .GAMMA. 2
> 1 , .kappa. .GAMMA. 1 .GAMMA. 2 > 2 , or .kappa. .GAMMA. 1
.GAMMA. 2 > 5. ##EQU00021##
In some such embodiments, D/L.sub.2 may be as large as 2, as large
as 3, as large as 5, as large as 7, or as large as 10.
[0061] 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 of the second resonator structure. In
some embodiments, D is also greater than one or more of: a
characteristic size L.sub.1 of the first resonator structure, a
characteristic thickness T.sub.1 of the first resonator structure,
and a characteristic width W.sub.1 of the first resonator
structure. The energy transfer is mediated by evanescent-tail
coupling of a resonant field of the first resonator structure and a
resonant field of the second resonator structure.
[0062] 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.
[0063] In some embodiments, the first resonator structure has a
resonant angular frequency .omega..sub.1, a Q-factor Q.sub.1, and a
resonance width .GAMMA..sub.1, the second resonator structure has a
resonant angular frequency .omega..sub.2, a Q-factor Q.sub.2, and a
resonance width .GAMMA..sub.2, and the energy transfer has a rate
.kappa.. In some embodiments, the absolute value of the difference
of the angular frequencies .omega..sub.1 and .omega..sub.2 is
smaller than the broader of the resonant widths .GAMMA..sub.1 and
.GAMMA..sub.2.
[0064] In some embodiments, the coupling to loss ratio
.kappa. .GAMMA. 1 .GAMMA. 2 > 0.5 , .kappa. .GAMMA. 1 .GAMMA. 2
> 1 , .kappa. .GAMMA. 1 .GAMMA. 2 > 2 , or .kappa. .GAMMA. 1
.GAMMA. 2 > 5. ##EQU00022##
In some such embodiments, D/L.sub.2 may be as large as 2, as large
as 3, as large as 5, as large as 7, or as large as 10.
[0065] 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 of the second resonator structure. In
some embodiments, D is also greater than one or more of: a
characteristic size L.sub.1 of the first resonator structure, a
characteristic thickness T.sub.1 of the first resonator structure,
and a characteristic width W of the first resonator structure. The
energy transfer is mediated by evanescent-tail coupling of a
resonant field of the first resonator structure and a resonant
field of the second resonator structure.
[0066] 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.
[0067] In some embodiments, the first resonator structure has a
resonant angular frequency .omega..sub.1, a Q-factor Q.sub.1, and a
resonance width .GAMMA..sub.1, the second resonator structure has a
resonant angular frequency .omega..sub.2, a Q-factor Q.sub.2, and a
resonance width .GAMMA..sub.2, and the energy transfer has a rate
.kappa.. In some embodiments, the absolute value of the difference
of the angular frequencies .omega..sub.1 and .omega..sub.2 is
smaller than the broader of the resonant widths .GAMMA..sub.1 and
.GAMMA..sub.2.
[0068] In some embodiments, the coupling to loss ratio
.kappa. .GAMMA. 1 .GAMMA. 2 > 0.5 , .kappa. .GAMMA. 1 .GAMMA. 2
> 1 , .kappa. .GAMMA. 1 .GAMMA. 2 > 2 , or .kappa. .GAMMA. 1
.GAMMA. 2 > 5. ##EQU00023##
In some such embodiments, D/L.sub.2 may be as large as 2, as large
as 3, as large as 5, as large as 7, or as large as 10.
[0069] 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.
[0070] 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.
[0071] As described in detail below, non-radiative energy transfer
refers to energy transfer effected primarily through the localized
near field, and, at most, secondarily through the radiative portion
of the field.
[0072] It is to be understood that an evanescent tail of a resonant
object is the decaying non-radiative 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.
[0073] The optimum efficiency frequency of a wireless energy
transfer system is the frequency at which the figure of merit
.kappa. .GAMMA. 1 .GAMMA. 2 ##EQU00024##
is maximized, all other factors held constant.
[0074] The resonant width (.GAMMA.) refers to the width of an
object's resonance due to object's intrinsic losses (e.g. loss to
absorption, radiation, etc.).
[0075] It is to be understood that a Q-factor (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 co and resonant width .GAMMA., the Q-factor
Q=.omega./2.GAMMA..
[0076] The energy transfer rate (K) 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.
[0077] It is to be understood that
Q.sub..kappa.=.omega./2.kappa..
[0078] 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.
[0079] 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.
[0080] Other features, objects, and advantages of the disclosure
will be apparent from the following detailed description.
BRIEF DESCRIPTION OF THE DRAWINGS
[0081] FIG. 1 shows a schematic of a wireless energy transfer
scheme.
[0082] FIG. 2 shows an example of a self-resonant conducting-wire
coil.
[0083] FIG. 3 shows a wireless energy transfer scheme featuring two
self-resonant conducting-wire coils
[0084] FIG. 4 shows an example of a capacitively loaded
conducting-wire coil, and illustrates the surrounding field.
[0085] FIG. 5 shows a wireless energy transfer scheme featuring two
capacitively loaded conducting-wire coils, and illustrates the
surrounding field.
[0086] FIG. 6 shows an example of a resonant dielectric disk, and
illustrates the surrounding field.
[0087] FIG. 7 shows a wireless energy transfer scheme featuring two
resonant dielectric disks, and illustrates the surrounding
field.
[0088] FIGS. 8a and 8b show schematics for frequency control
mechanisms.
[0089] FIGS. 9a through 9c illustrate a wireless energy transfer
scheme in the presence of various extraneous objects.
[0090] FIG. 10 illustrates a circuit model for wireless energy
transfer.
[0091] FIG. 11 illustrates the efficiency of a wireless energy
transfer scheme.
[0092] FIG. 12 illustrates parametric dependences of a wireless
energy transfer scheme.
[0093] FIG. 13 plots the parametric dependences of a wireless
energy transfer scheme.
[0094] FIG. 14 is a schematic of an experimental system
demonstrating wireless energy transfer.
[0095] FIGS. 15-17. Plot experiment results for the system shown
schematically in FIG. 14.
DETAILED DESCRIPTION
[0096] FIG. 1 shows a schematic that generally describes one
embodiment of the invention, in which energy is transferred
wirelessly between two resonant objects.
[0097] Referring to FIG. 1, energy is transferred, over a distance
D, between a resonant source object having a characteristic size
L.sub.1 and a resonant device object of characteristic size
L.sub.2. 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 non-radiatively from the source
object to the device object, and consumed by the power consuming
device. The wireless non-radiative 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.
[0098] 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. Similarly, in some
embodiments multiple sources can transfer energy to one or more
device objects. For example, as explained at in the paragraph
bridging pages 4-5 of U.S. Provisional Application No. 60/698,442
to which the present application claims benefit and which is
incorporated by reference above, for certain applications having
uneven power transfer to the device object as the distance between
the device and the source changes, multiple sources can be
strategically placed to alleviate the problem, and/or the peak
amplitude of the source can be dynamically adjusted. Furthermore,
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.
[0099] Initially, we present a theoretical framework for
understanding non-radiative wireless energy transfer. Note however
that it is to be understood that the scope of the invention is not
bound by theory.
[0100] Coupled Mode Theory
[0101] An appropriate analytical framework for modeling the
resonant energy-exchange between two resonant objects 1 and 2 is
that of "coupled-mode theory" (CMT). The field of the system of two
resonant objects 1 and 2 is approximated by
F(r,t).apprxeq.a.sub.1(t)F.sub.1(r)+a.sub.2(t)F.sub.2(r), where
F.sub.1,2(r) are the eigenmodes of 1 and 2 alone, normalized to
unity energy, and the field amplitudes a.sub.1,2(t) are defined so
that |a.sub.1,2(t)|.sup.2 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:
da 1 dt = - i ( .omega. 1 - i .GAMMA. 1 ) a 1 + i .kappa. a 2 da 2
dt = - i ( .omega. 2 - i .GAMMA. 2 ) a 2 + i .kappa. a 1 , ( 1 )
##EQU00025##
where .omega..sub.1,2 are the individual angular eigenfrequencies
of the eigenmodes, .GAMMA..sub.1,2 are the resonance widths due to
the objects' intrinsic (absorption, radiation etc.) losses, and
.kappa. is the coupling coefficient. Eqs. (1) show that at exact
resonance (.omega..sub.1=.omega..sub.2 and
.GAMMA..sub.1=.omega..sub.2), the eigenmodes of the combined system
are split by 2.kappa.; the energy exchange between the two objects
takes place in time .about..pi./2.kappa. and is nearly perfect,
apart for losses, which are minimal when the coupling rate is much
faster than all loss rates (.kappa.>>.GAMMA..sub.1,2). The
coupling to loss ratio .kappa./ {square root over
(.GAMMA..sub.1.GAMMA..sub.2)} serves as a figure-of-merit in
evaluating a system used for wireless energy-transfer, along with
the distance over which this ratio can be achieved. The regime
.kappa./ {square root over (.GAMMA..sub.1.GAMMA..sub.2)}>>1
is called "strong-coupling" regime.
[0102] In some embodiments, the energy-transfer application
preferably uses resonant modes of high Q=.omega./2.delta.,
corresponding to low (i.e. slow) intrinsic-loss rates .GAMMA.. This
condition may be satisfied where the coupling is implemented using,
not the lossy radiative far-field, but the evanescent (non-lossy)
stationary near-field.
[0103] To implement an energy-transfer 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=.omega..sup.2/c.sup.2
where {right arrow over (k)} is the wave vector, .omega. 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 long-lived (so-called "high-Q") states can be found, whose
tails display the needed exponential or exponential-like 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 energy-transfer 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.
[0104] Furthermore, in some embodiments, small
Q.sub..kappa.=.omega./2.kappa. corresponding to strong (i.e. fast)
coupling rate .kappa. is preferred over distances larger than the
characteristic sizes of the objects. Therefore, since the extent of
the near-field into the area surrounding a finite-sized resonant
object is set typically by the wavelength, in some embodiments,
this mid-range non-radiative coupling can be achieved using
resonant objects of subwavelength size, and thus significantly
longer evanescent field-tails. 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
possibly-mobile resonant device-object. Note, though, that in some
embodiments, the resonant source-object will be immobile and thus
less restricted in its allowed geometry and size, which can be
therefore chosen large enough that the near-field 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.
[0105] 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 .omega..sub.1,2,
Q.sub.1,2 and Q.sub..kappa. described above and how to choose these
parameters for particular embodiments in order to produce a
desirable figure-of-merit .kappa./ {square root over
(.GAMMA..sub.1.GAMMA..sub.2)}= {square root over
(Q.sub.1Q.sub.2)}/Q.sub..kappa.. In particular, this figure of
merit is typically maximized when .omega..sub.1,2 are tuned to a
particular angular frequency {tilde over (.omega.)}, thus, if
{tilde over (F)} is half the angular-frequency width for which
{square root over (.GAMMA..sub.1.GAMMA..sub.2)}= {square root over
(Q.sub.1Q.sub.2)}/Q.sub..kappa. is above half its maximum value at
{tilde over (.omega.)} the angular eigenfrequencies .omega..sub.1,2
should typically be tuned to be close to {tilde over (.omega.)} to
within the width {tilde over (.GAMMA.)}.
[0106] In addition, as described below, Q.sub.1,2 can sometimes be
limited not from intrinsic loss mechanisms but from external
perturbations. In those cases, producing a desirable
figure-of-merit translates to reducing Q.sub..kappa. (i.e.
increasing the coupling). Accordingly we will demonstrate how, for
particular embodiments, to reduce Q.sub..kappa..
[0107] Self-Resonant Conducting Coils
[0108] In some embodiments, one or more of the resonant objects are
self-resonant conducting coils. Referring to FIG. 2, a conducting
wire of length l and cross-sectional radius a is wound into a
helical coil of radius r and height h (namely with N= {square root
over (l.sup.2-h.sup.2)}/2.pi.r 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 .omega.. The nature of the resonance lies in
the periodic exchange of energy from the electric field within the
capacitance of the coil, due to the charge distribution .rho.(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 .gradient.j=i.omega..rho. implies that: (i)
this periodic exchange is accompanied by a .pi./2 phase-shift
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 .rho..sub.1(x) and I(x)
are respectively the linear charge and current densities in the
wire, where x runs along the wire,
q.sub.o1/2=.intg.dx|.rho..sub.1(x)| 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=max{|I(x)|} is the maximum
positive value of the linear current distribution, then
I.sub.o=.omega.q.sub.o. 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:
U .ident. 1 2 I o 2 L L = .mu. o 4 .pi. I o 2 .intg. .intg. dxdx '
j ( x ) j ( x ' ) x - x ' , ( 2 ) U .ident. 1 2 q o 2 1 C 1 C = 1 4
.pi. o q o 2 .intg. .intg. dxdx ' .rho. ( x ) .rho. ( x ' ) x - x '
, ( 3 ) ##EQU00026##
where .mu..sub.0 and .epsilon..sub.0 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 .omega.=1/ {square root over (LC)} and
Z= {square root over (L/C)} respectively.
[0109] 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
from the amount of power absorbed inside the wire and a total
radiation resistance R.sub.rad from the amount of power radiated
due to electric- and magnetic-dipole radiation:
P abs .ident. 1 2 I o 2 R abs R abs .apprxeq. .zeta. c l 2 .pi. a I
rms 2 I o 2 ( 4 ) P rad .ident. 1 2 I o 2 R rad R rad .apprxeq.
.zeta. o 6 .pi. [ ( .omega. p c ) 2 + ( .omega. m c ) 4 ] , ( 5 )
##EQU00027##
where c=1 {square root over (.mu..sub.o.epsilon..sub.o)} and
.zeta..sub.o= {square root over (.mu..sub.o.epsilon..sub.o)} so are
the light velocity and light impedance in free space, the impedance
.zeta..sub.c is .zeta..sub.c=1/.sigma..delta.= {square root over
(.mu..sub.o.omega./2.sigma.)} with a the conductivity of the
conductor and .delta. the skin depth at the frequency .omega.,
I rms 2 = 1 l .intg. dx I ( x ) 2 , p = .intg. dxr .rho. l ( x )
##EQU00028##
is the electric-dipole moment of the coil and
m=1/2.intg.dxr.times.j(x) is the magnetic-dipole moment of the
coil. For the radiation resistance formula Eq. (5), the assumption
of operation in the quasi-static regime
(h,r<<.lamda.=2.pi.c/.omega.) 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=Z/R.sub.abs and
Q.sup.rad=Z/R.sub.rad respectively.
[0110] 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 conducting-wire 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
transmission-line-like model, and solve it by brute force. We
instead present a model that is (as described below) in good
agreement (.about.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
cos(.pi.x/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 .rho..sub.l(x)=.rho..sub.o sin
(.pi.x/l), and thus
q.sub.o=.intg..sub.0.sup.1/2dx.rho..sub.o|sin(.pi.x/l)|=.rho..sub.ol/.pi.-
. Using these sinusoidal profiles we find the so-called
"self-inductance" L.sub.s and "self-capacitance" C.sub.s of the
coil by computing numerically the integrals Eq. (2) and (3); the
associated frequency and effective impedance are .omega..sub.s and
Z.sub.s respectively. The "self-resistances" R.sub.s are given
analytically by Eq. (4) and (5) using
I rms 2 = 1 l .intg. - l / 2 l / 2 dx I o cos ( .pi. x / l ) 2 = 1
2 I o 2 , p = q o ( 2 .pi. h ) 2 + ( 4 N cos ( .pi. N ) ( 4 N 2 - 1
) .pi. r ) 2 ##EQU00029## and ##EQU00029.2## m = I o ( 2 .pi. N
.pi. r 2 ) 2 + ( cos ( .pi. N ) ( 12 N 2 - 1 ) - sin ( .pi. N )
.pi. N ( 4 N 2 - 1 ) ( 16 N 4 - 8 N 2 + 1 ) .pi. hr ) 2 ,
##EQU00029.3##
and therefore the associated Q.sub.s factors may be calculated.
[0111] The results for two particular embodiments of resonant coils
with subwavelength modes of .lamda..sub.s/r.gtoreq.70 (i.e. those
highly suitable for near-field coupling and well within the
quasi-static 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 subwavelength-coil resonant
modes. Note that, for conducting material, copper (.sigma.=5.99810
-7 S/m) was used. It can be seen that expected quality factors at
microwave frequencies are Q.sub.s.sup.abs.gtoreq.1000 and
Q.sub.s.sup.ad.gtoreq.5000.
TABLE-US-00001 TABLE 1 single coil .lamda..sub.s/r f (MHz)
Q.sub.s.sup.rad Q.sub.s.sup.abs Q.sub.s =
.omega..sub.s/2.GAMMA..sub.s r = 30 cm, h = 20 cm, a = 1 cm, N = 4
74.7 13.39 4164 8170 2758 r = 10 cm, h = 3 cm, a = 2 mm, N = 6 140
21.38 43919 3968 3639
[0112] Referring to FIG. 3, in some embodiments, energy is
transferred between two self-resonant conducting-wire coils. The
electric and magnetic fields are used to couple the different
resonant conducting-wire 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
.rho..sub.1,2(x) and j.sub.1,2 (x), total charges and peak currents
respectively as q.sub.1,2 and I.sub.1,2, and capacitances and
inductances respectively as C.sub.1,2 and L.sub.1,2, which are the
analogs of .rho.(x), j(x), q.sub.o, I.sub.o, C and L for the
single-coil case and are therefore well defined, we can define
their mutual capacitance and inductance through the total
energy:
U .ident. U 1 + U 2 + 1 2 ( q 1 * q 2 + q 2 * q 1 ) / M C + 1 2 ( I
1 * I 2 + I 2 * I 1 ) M L 1 / M C = 1 4 .pi. o q 1 q 2 .intg.
.intg. dxdx ' .rho. 1 ( x ) .rho. 2 ( x ' ) x - x ' u , M L = .mu.
o 4 .pi. I 1 I 2 .intg. .intg. dxdx ' j 1 ( x ) j 2 ( x ' ) x - x '
u , ( 6 ) ##EQU00030##
where U.sub.1=1/2q.sub.1.sup.2/C.sub.1=1/2I.sub.1.sup.2 L.sub.1,
U.sub.2=1/2q.sub.2.sup.2/C.sub.2=1/2I.sub.2.sup.2 L.sub.2 and the
retardation factor of u=exp(i.omega.|x-x'|/c) inside the integral
can been ignored in the quasi-static regime D<<.lamda. of
interest, where each coil is within the near field of the other.
With this definition, the coupling coefficient is given by
.kappa.=.omega. {square root over
(C.sub.1C.sub.2)}/2M.sub.C+.omega.M.sub.L/2 {square root over
(L.sub.1L.sub.2)}.revreaction.Q.sub..kappa..sup.-1=(M.sub.C/
{square root over (C.sub.1C.sub.2)}).sup.-1+( {square root over
(L.sub.1L.sub.2)}/M.sub.L).sup.-1.
[0113] Therefore, to calculate the coupling rate between two
self-resonant 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 and
inductance M.sub.L,s between two self-resonant coils at a distance
D between their centers, and thus Q.sub..kappa.,s is also
determined.
TABLE-US-00002 TABLE 2 pair of coils D/r Q = .omega./2.GAMMA.
Q.sub..kappa. = .omega./2.kappa. .kappa./.GAMMA. r = 30 cm, h = 20
cm, 3 2758 38.9 70.9 a = 1 cm, N = 4 5 2758 139.4 19.8 .lamda./r
.apprxeq. 75 7 2758 333.0 8.3 Q.sub.s.sup.abs .apprxeq. 8170,
Q.sub.s.sup.rad .apprxeq. 4164 10 2758 818.9 3.4 r = 10 cm, h = 3
cm, 3 3639 61.4 59.3 a = 2 mm, N = 6 5 3639 232.5 15.7 .lamda./r
.apprxeq. 140 7 3639 587.5 6.2 Q.sub.s.sup.abs .apprxeq. 3968,
Q.sub.s.sup.rad .apprxeq. 43919 10 3639 1580 2.3
[0114] 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 figure-of-merit 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 coupling-to-loss ratios are in the range
.kappa./.GAMMA..about.2-70.
Capacitively-Loaded Conducting Loops or Coils
[0115] In some embodiments, one or more of the resonant objects are
capacitively-loaded 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 .epsilon., and everything is surrounded by air (as
shown, N=1 and h=0). The plates have a capacitance
C.sub.p=.epsilon..sub.0.epsilon.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
and C.sub.s, which are calculated for a self-resonant 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 .rho. inside the wire is reduced, so
C<C.sub.s, and thus, from the charge conservation equation, the
current distribution j flattens out, so L>L.sub.s. The resonant
frequency for this system is .omega.=1/ {square root over
(L(C+C.sub.p))}<.omega..sub.s=1/ {square root over
(L.sub.sC.sub.s)}, and I(x).fwdarw.I.sub.o
cos(.pi.x/l)C.fwdarw.C.sub.s.omega..fwdarw..omega..sub.s, as
C.sub.p.fwdarw.0.
[0116] 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>>C.sub.s>C, then .omega..apprxeq.1/
{square root over (LC.sub.p)}<<.omega..sub.s and Z.apprxeq.
{square root over (L/C.sub.p)}<<Z.sub.s, 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=.mu..sub.or[ln (8r/a)-2]N.sup.2. Explicit
analytical formulas are again available for R from Eq. (4) and (5),
since I.sub.rms=I.sub.o, |p|.apprxeq.0 and |m|=I.sub.oN.pi.r.sup.2
(namely only the magnetic-dipole term is contributing to
radiation), so we can determine also Q.sup.abs=.omega.L/R.sub.abs
and Q.sup.rad=.omega.L/R.sub.rad. 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>>C.sub.s .revreaction..omega.<<.omega..sub.s,
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 self-capacitance C.sub.s to begin
with, which usually holds, for the types of coils under
consideration, when N=1, so that the self-capacitance 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 self-capacitance comes from the charge distribution across
adjacent turns, which is small if the separation between adjacent
turns is large.
[0117] The external loading capacitance C.sub.p 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=.omega.L/(R.sub.abs+R.sub.rad)
becomes highest at the optimal frequency
.omega. ~ = [ c 4 .pi. o 2 .sigma. 1 a Nr 3 ] 2 7 , ( 7 )
##EQU00031##
reaching the value
Q ~ = 6 7 .pi. ( 2 .pi. 2 .eta. o .sigma. a 2 N 2 r ) 3 7 [ ln ( 8
r a ) - 2 ] . ( 8 ) ##EQU00032##
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 c<<co 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 self-capacitance. A coil
with large h can be used, if the self-capacitance needs to be
reduced compared to the external capacitance, but then the formulas
for L and {tilde over (.omega.)}, {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.
[0118] The results of the above analysis for two embodiments of
subwavelength modes of .pi./r.gtoreq.70 (namely highly suitable for
near-field coupling and well within the quasi-static 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 constant-current
assumption and the resulting analytical formulas, mode-solving
calculations were also performed using another completely
independent method: computational 3D finite-element
frequency-domain (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 .zeta..sub.c=
{square root over (.mu..sub.o.omega./2.sigma.)} boundary condition,
valid as long as .zeta..sub.c/.zeta..sub.c<<1 (<10.sup.-5
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
subwavelength-loop resonant modes. Note that for conducting
material copper (.alpha.=5.998-10.sup.7S/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 low-MHz microwave range and the expected
quality factors are Q.sup.abs.gtoreq.1000 and
Q.sup.rad.gtoreq.10000.
TABLE-US-00003 TABLE 3 single coil .lamda./r f (MHz) Q.sup.rad
Q.sup.abs Q = .omega./2.GAMMA. r = 30 cm, a = 2 cm 111.4 (112.4)
8.976 (8.897) 29546 (30512) 4886 (5117) 4193 (4381) .epsilon. = 10,
A = 138 cm.sup.2, d = 4 mm r = 10 cm, a = 2 mm 69.7 (70.4) 43.04
(42.61) 10702 (10727) 1545 (1604) 1350 (1395) .epsilon. = 10, A =
3.14 cm.sup.2, d = 1 mm
[0119] Referring to FIG. 5, in some embodiments, energy is
transferred between two capacitively-loaded coils. For the rate of
energy transfer between two capacitively-loaded coils 1 and 2 at
distance D between their centers, the mutual inductance M.sub.L can
be evaluated numerically from Eq. (6) by using constant current
distributions in the case .omega.<<.omega..sub.s. In the case
h=0, the coupling is only magnetic and again we have an analytical
formula, which, in the quasi-static limit r<<D<<.lamda.
and for the relative orientation shown in FIG. 4, is
M.sub.L.apprxeq..pi..mu..sub.o/2(r.sub.1r.sub.2).sup.2N.sub.1N.sub.2/D.su-
p.3, which means that Q.sub..kappa..varies.(D/ {square root over
(r.sub.1r.sub.2)}).sup.3 is independent of the frequency co and the
number of turns N.sub.1, N.sub.2. Consequently, the resultant
coupling figure-of-merit of interest is
.kappa. .GAMMA. 1 .GAMMA. 2 = Q 1 Q 2 Q .kappa. .apprxeq. ( r 1 r 2
D ) 3 .pi. 2 .eta. o r 1 r 2 .lamda. N 1 N 2 j = 1 , 2 ( .pi. .eta.
o .lamda. .sigma. r j a j N j + 8 3 .pi. 5 .eta. o ( r j .lamda. )
4 N j 2 ) 1 / 2 , ( 9 ) ##EQU00033##
which again is more accurate for N.sub.1=N.sub.2=1.
[0120] From Eq. (9) it can be seen that the optimal frequency
{tilde over (.omega.)}, where the figure-of-merit is maximized to
the value , is that where {square root over (Q.sub.1Q.sub.2)} is
maximized, since Q.sub..kappa. does not depend on frequency (at
least for the distances D<<.lamda. of interest for which the
quasi-static 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 single-coil Q.sub.1
and Q.sub.2 peak. For same coils, it is given by Eq. (7) and then
the figure-of-merit Eq. (9) becomes
= Q ~ Q .kappa. .apprxeq. ( r D ) 3 3 7 ( 2 .pi. 2 .eta. o .sigma.
a 2 N 2 r ) 3 7 . ( 10 ) ##EQU00034##
Typically, one should tune the capacitively-loaded conducting loops
or coils, so that their angular eigenfrequencies are close to
{tilde over (.omega.)} within {tilde over (.GAMMA.)}, which is half
the angular frequency width for which {square root over
(Q.sub.1Q.sub.2)}/Q.sub..kappa.>/2.
[0121] 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 figure-of-merit
.kappa./.GAMMA. as a function of the coupling distance D, for the
two cases. Note that the average numerical .GAMMA..sup.rad shown
are again slightly different from the single-loop value of FIG. 3,
analytical results for .GAMMA..sup.rad are not shown but the
single-loop 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 constant-current assumption a good
one and computed M.sub.L numerically from Eq. (6). Indeed the
accuracy can be confirmed by their agreement with the computational
FEFD mode-solver simulations, which give .kappa. through the
frequency splitting (=2.kappa.) of the two normal modes of the
combined system. The results show that for medium distances
D/r=10-3 the expected coupling-to-loss ratios are in the range
.kappa./.GAMMA..about.0.5-50.
TABLE-US-00004 TABLE 4 pair of coils D/r Q.sup.rad Q =
.omega./2.GAMMA. Q.sub..kappa. = .omega./2.kappa. .kappa./.GAMMA. r
= 30 cm, a = 2 cm 3 30729 4216 62.6 (63.7) 67.4 (68.7) .epsilon. =
10, A = 138 cm.sup.2, d = 4 mm 5 29577 4194 235 (248) 17.8 (17.6)
.lamda./r .apprxeq. 112 7 29128 4185 589 (646) 7.1 (6.8) Q.sup.abs
.apprxeq. 4886 10 28833 4177 1539 (1828) 2.7 (2.4) r = 10 cm, a = 2
mm 3 10955 1355 85.4 (91.3) 15.9 (15.3) .epsilon. = 10, A = 3.14
cm.sup.2, d = 1 mm 5 10740 1351 313 (356) 4.32 (3.92) .lamda./r
.apprxeq. 70 7 10759 1351 754 (925) 1.79 (1.51) Q.sup.abs .apprxeq.
1546 10 10756 1351 1895 (2617) 0.71 (0.53)
Optimization of {square root over
(Q.sub.1Q.sub.2)}/Q.sub..kappa.
[0122] In some embodiments, the results above can be used to
increase or optimize the performance of a wireless energy transfer
system which employs capacitively-loaded coils. For example, the
scaling of Eq. (10) with the different system parameters one sees
that to maximize the system figure-of-merit .kappa./.GAMMA. one
can, for example: [0123] 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.
[0124] 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 cross-sectional 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. [0125] 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. [0126] For fixed desired distance vs. loop-size
ratio D/r, decrease the radius of the loop r. In typical
embodiments, this action is limited by physical size
considerations. [0127] 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 coupling-to-loss 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.
[0128] Adjust the alignment and orientation between the two coils.
The figure-of-merit 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. [0129] 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.
[0130] 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 single-turn loops with a given radius in order to achieve
a specific performance in terms of .kappa./.GAMMA. at a given D/r
between them, when the material is copper
(.sigma.=5.99810.sup.7S/m):
D/r=5,.kappa./.GAMMA..gtoreq.10,r=30 cma.gtoreq.9 mm
D/r=5,.kappa./.GAMMA..gtoreq.10,r=5 cma.gtoreq.3.7 mm
D/r=5,.kappa./.GAMMA.>20,r=30 cma.gtoreq.20 mm
D/r=5,.kappa./.GAMMA..gtoreq.20,r=5 cma.gtoreq.8.3 mm
D/r=10,.kappa./.GAMMA..gtoreq.1,r=30 cma.gtoreq.7 mm
D/r=10,.kappa./.GAMMA..gtoreq.1,r=5 cma.gtoreq.2.8 mm
D/r=10,.kappa./.GAMMA..gtoreq.3,r=30 cma.gtoreq.25 mm
D/r=10,.kappa./.GAMMA.>3,r=5 cma.gtoreq.10 mm
[0131] 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, h.sub.d, a.sub.d,
N.sub.d) are very restricted. However, in some such embodiments,
the restrictions on the source object (r.sub.s, h.sub.s, a.sub.s,
N.sub.s) 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.about.1 m for charging a laptop on a table wirelessly from
the floor). Listed below are examples (simplified to the case
N.sub.s=N.sub.d=1 and h.sub.s=h.sub.d=0) of how one can vary the
dimensions of the source object to achieve the desired system
performance in terms of .kappa./ {square root over
(.GAMMA..sub.s.GAMMA..sub.d)}, when the material is again copper
(.alpha.=5.99810.sup.7S/m):
D=1.5 m,.kappa./ {square root over
(.GAMMA..sub.s.GAMMA..sub.d)}.gtoreq.15,r.sub.d=30 cm,a.sub.d=6
mmr.sub.s=1.158 m,a.sub.s.gtoreq.5 mm
D=1.5 m,.kappa./ {square root over
(.GAMMA..sub.s.GAMMA..sub.d)}.gtoreq.30,r.sub.d=30 cm,a.sub.d=6
mmr.sub.s=1.15 m,a.sub.s.gtoreq.33 mm
D=1.5 m,.kappa./ {square root over
(.GAMMA..sub.s.GAMMA..sub.d)}.gtoreq.1,r.sub.d=5 cm,a.sub.d=4
mmr.sub.s=1.119 m,a.sub.s.gtoreq.7 mm
D=1.5 m,.kappa./ {square root over
(.GAMMA..sub.s.GAMMA..sub.d)}.gtoreq.2,r.sub.d=5 cm,a.sub.d=4
mmr.sub.s=1.119 m,a.sub.s.gtoreq.52 mm
D=2 m,.kappa./ {square root over
(.GAMMA..sub.s.GAMMA..sub.d)}.gtoreq.10,r.sub.d=30 cm,a.sub.d=6
mmr.sub.s=1.518 m,a.sub.s.gtoreq.7 mm
D=2 m,.kappa./ {square root over
(.GAMMA..sub.s.GAMMA..sub.d)}.gtoreq.20,r.sub.d=30 cm,a.sub.d=6
mmr.sub.s=1.514 m,a.sub.s.gtoreq.50 mm
D=2 m,.kappa./ {square root over
(.GAMMA..sub.s.GAMMA..sub.d)}.gtoreq.0.5,r.sub.d=5 cm,a.sub.d=4
mmr.sub.s=1.491 m,a.sub.s.gtoreq.5 mm
D=2 m,.kappa./ {square root over
(.GAMMA..sub.s.GAMMA..sub.d)}.gtoreq.1,r.sub.d=5 cm,a.sub.d=4
mmr.sub.s=1.491 m,a.sub.s.gtoreq.36 mm
[0132] Optimization of Q.sub..kappa.
[0133] 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 figure-of-merit by decreasing Q.sub..kappa.
(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: [0134] Increase the wire radii a.sub.1 and a.sub.2.
In typical embodiments, this action is limited by physical size
considerations. [0135] For fixed desired distance D of energy
transfer, increase the radii of the coils r.sub.1 and r.sub.2. In
typical embodiments, this action is limited by physical size
considerations. [0136] For fixed desired distance vs. coil-sizes
ratio D/ {square root over (r.sub.1r.sub.2)}, only the weak
(logarithmic) dependence of the inductance remains, which suggests
that one should decrease the radii of the coils r.sub.1 and
r.sub.2. In typical embodiments, this action is limited by physical
size considerations. [0137] 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. [0138]
Finally, note that the heights of the coils h.sub.1 and h.sub.2 are
other available design parameters, which have an impact to the
coupling similar to that of their radii r.sub.1 and r.sub.2, and
thus the design rules are similar.
[0139] Further practical considerations apart from efficiency, e.g.
physical size limitations, will be discussed in detail below.
[0140] It is also important to appreciate the difference between
the above described resonant-coupling inductive scheme and the
well-known non-resonant 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 .about.Q.sup.2 (.about.10.sup.6) times more power delivered for
work at the device than the traditional non-resonant mechanism.
This is why only close-range contact-less medium-power (.about.W)
transfer is possible with the latter, while with resonance either
close-range but large-power (.about.kW) transfer is allowed or, as
currently proposed, if one also ensures operation in the
strongly-coupled regime, medium-range and medium-power transfer is
possible. Capacitively-loaded conducting loops are currently used
as resonant antennas (for example in cell phones), but those
operate in the far-field regime with D/r>>1,
r/.lamda..about.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.
[0141] Inductively-Loaded Conducting Rods
[0142] A straight conducting rod of length 2h and cross-sectional
radius a has distributed capacitance and distributed inductance,
and therefore it supports a resonant mode of angular frequency
.omega.. Using the same procedure as in the case of self-resonant
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
.omega.=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 cos(.pi.x/2h), since it has to be zero at the
open ends of the rod. This is the well-known half-wavelength
electric dipole resonant mode.
[0143] In some embodiments, one or more of the resonant objects are
inductively-loaded conducting rods. A straight conducting rod of
length 2h and cross-sectional 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 .mu., and everything is surrounded by air. The coil
has an inductance L.sub.c, which is added to the distributed
inductance of the rod and thus modifies its resonance. Note
however, that the presence of the center-loading 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
and C.sub.s, which are calculated for a self-resonant 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, and thus, from the charge
conservation equation, the linear charge distribution .rho..sub.l
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. The resonant frequency for
this system is .omega.=1/ {square root over
((L+L.sub.c)C)}<.omega..sub.s=1/ {square root over
(L.sub.sC.sub.s)}, and I(x).fwdarw.I.sub.o
cos(.pi.x/2h)L.fwdarw.L.sub.s.omega..fwdarw..omega..sub.s, as
L.sub.c.fwdarw.0.
[0144] 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>>L.sub.s>L, then .omega..ltoreq.1/
{square root over (L.sub.cC)}<<.omega., and Z.apprxeq.
{square root over (L.sub.c/C)}>>Z.sub.s, while the current
distribution is triangular along the rod (with maximum at the
center-loading 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/(.pi..epsilon..sub.oh)[ln(h/a)-1]. Explicit
analytical formulas are again available for R from Eq. (4) and (5),
since I.sub.rms=I.sub.o, |p|=q.sub.oh and |m|=0 (namely only the
electric-dipole term is contributing to radiation), so we can
determine also Q.sup.abs=1/.omega.CR.sub.abs and
Q.sup.rad=1/CR.sub.rad. 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>>L.sub.s .revreaction..omega.<<.omega..sub.s is
satisfied. This condition is relatively easily satisfied, since
typically a conducting rod has very small self-inductance L.sub.s
to begin with.
[0145] Another important loss factor in this case is the resistive
loss inside the coil of the external loading inductor L.sub.c 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 turns of conducting wire of cross-sectional
radius a.sub.Bc wrapped around a cylindrically symmetric coil
former, which forms a coil with a square cross-section of side
r.sub.Bc, where the inner side of the square is also at radius
r.sub.Bc (and thus the outer side of the square is at radius
2r.sub.Bc), therefore N.sub.Bc.apprxeq.(r.sub.Bc/2a.sub.Bc).sup.2.
The inductance of the coil is then
L.sub.c=2.0285.mu..sub.or.sub.BcN.sub.Bc.sup.2.apprxeq.2.0285.mu.-
.sub.or.sub.Bc.sup.5/8a.sub.Bc.sup.4 and its resistance
R C .apprxeq. 1 .sigma. l Bc .pi. a Bc 2 1 + .mu. o .omega. .sigma.
2 ( a Bc 2 ) 2 , ##EQU00035##
where the total wire length is
I.sub.Bc.apprxeq.2.pi.(3r.sub.Bc/2)N.sub.Bc.apprxeq.3.pi.r.sub.Bc.sup.3/4-
a.sub.Bc.sup.2 and we have used an approximate square-root law for
the transition of the resistance from the dc to the ac limit as the
skin depth varies with frequency.
[0146] The external loading inductance L.sub.c provides the freedom
to tune the resonant frequency. (For example, for a Brooks coil
with a fixed size r.sub.Bc, the resonant frequency can be reduced
by increasing the number of turns N.sub.Bc by decreasing the wire
cross-sectional radius a.sub.Bc. Then the desired resonant angular
frequency .omega.=1/ {square root over (L.sub.cC)} is achieved for
a.sub.Bc.apprxeq.(2.0285.mu..sub.or.sub.Bc.sup.5.omega..sup.2C).sup.1/4
and the resulting coil quality factor is
Q.sub.c.apprxeq.0.169.mu..sub.o.sigma.r.sub.Bc.sup.2.omega./
{square root over (1+.omega..sup.2.sigma. {square root over
(2.0285.mu..sub.o(r.sub.Bc/4).sup.5C)})}). Then, for the particular
simple case L.sub.c>>L.sub.s, for which we have analytical
formulas, the total Q=1/.omega.C(R.sub.c+R.sub.abs+R.sub.rad)
becomes highest at some optimal frequency {tilde over (.omega.)},
reaching the value {tilde over (Q)}, both determined by the
loading-inductor specific design. (For example, for the Brooks-coil
procedure described above, at the optimal frequency {tilde over
(Q)}.apprxeq.Q.sub.c.apprxeq.0.8
(.mu..sub.o.sigma..sup.2r.sub.Bc.sup.3/C).sup.1/4) 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
(.omega.)}<<.omega..sub.s and, as explained above, this is
easy to design for by using a large inductance.
[0147] The results of the above analysis for two embodiments, using
Brooks coils, of subwavelength modes of .lamda./h>200 (namely
highly suitable for near-field coupling and well within the
quasi-static limit) at the optimal frequency {tilde over (.omega.)}
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 subwavelength-loop resonant modes. Note that for
conducting material copper (.sigma.=5.99810.sup.7S/m) was used. The
results show that, in some embodiments, the optimal frequency is in
the low-MHz microwave range and the expected quality factors are
Q.sup.abs.gtoreq.1000 and Q.sup.rad.gtoreq.100000.
TABLE-US-00005 TABLE 5 single rod .lamda./h f (MHz) Q.sup.rad
Q.sup.abs Q = .omega./2.GAMMA. h = 30 cm, a = 2 cm (403.8) (2.477)
(2.72*10.sup.6) (7400) (7380) .mu. = 1, r.sub.Bc = 2 cm, a.sub.Bc =
0.88 mm, N.sub.Bc = 129 h = 10 cm, a = 2 mm (214.2) (14.010)
(6.92*10.sup.5) (3908) (3886) .mu. = 1, r.sub.Bc = 5 mm, a.sub.Bc =
0.25 mm,
[0148] In some embodiments, energy is transferred between two
inductively-loaded rods. For the rate of energy transfer between
two inductively-loaded rods 1 and 2 at distance D between their
centers, the mutual capacitance M.sub.C can be evaluated
numerically from Eq. (6) by using triangular current distributions
in the case .omega.<<.omega..sub.s. In this case, the
coupling is only electric and again we have an analytical formula,
which, in the quasi-static limit h<<D<<.lamda. and for
the relative orientation such that the two rods are aligned on the
same axis, is
1/M.sub.C.apprxeq.1/2.pi..epsilon..sub.o(h.sub.1h.sub.2).sup.2/D.sup.3,
which means that Q.sub..kappa..varies.(D/ {square root over
(h.sub.1h.sub.2)}).sup.3 is independent of the frequency .omega..
Consequently, one can get the resultant coupling figure-of-merit of
interest
.kappa. .GAMMA. 1 .GAMMA. 2 = Q 1 Q 2 Q .kappa. . ##EQU00036##
It can be seen that the optimal frequency {tilde over (.omega.)},
where the figure-of-merit is maximized to the value , is that where
Q.sub.1Q.sub.2 is maximized, since Q.sub..kappa. does not depend on
frequency (at least for the distances D<<.lamda. of interest
for which the quasi-static 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
single-rod Q.sub.1 and Q.sub.2 peak. Typically, one should tune the
inductively-loaded conducting rods, so that their angular
eigenfrequencies are close to {tilde over (.omega.)} within {tilde
over (.GAMMA.)}, which is half the angular frequency width for
which {square root over (Q.sub.1Q.sub.2)}/Q.sub..kappa.>/2.
[0149] 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
figure-of-merit .kappa./.GAMMA. as a function of the coupling
distance D, for the two cases. Note that for .GAMMA..sup.rad the
single-rod value is used. Again we chose L.sub.c>>L.sub.s to
make the triangular-current assumption a good one and computed
M.sub.C numerically from Eq. (6). The results show that for medium
distances D/h=10-3 the expected coupling-to-loss ratios are in the
range .kappa./.GAMMA..about.0.5-100.
TABLE-US-00006 TABLE 6 pair of rods D/h Q.kappa. = .omega./2.kappa.
.kappa./.GAMMA. h = 30 cm, a = 2 cm 3 (70.3) (105.0) .mu. = 1,
r.sub.Bc = 2 cm, a.sub.Bc = 0.88 mm, N.sub.Bc = 129 5 (389) (19.0)
.lamda./h .apprxeq. 404 7 (1115) (6.62) Q .apprxeq. 7380 10 (3321)
(2.22) h = 10 cm, a = 2 mm 3 (120) (32.4) .mu. = 1, r.sub.Bc = 5
mm, a.sub.Bc = 0.25 mm, N.sub.Bc = 103 5 (664) (5.85) .lamda./h
.apprxeq. 214 7 (1900) (2.05) Q .apprxeq. 3886 10 (5656) (0.69)
[0150] Dielectric Disks
[0151] 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 e surrounded by air that supports high-Q
"whispering-gallery" 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. High-Q.sub.rad and
long-tailed subwavelength resonances can be achieved when the
dielectric permittivity e is large and the azimuthal field
variations are slow (namely of small principal number m). Material
absorption is related to the material loss tangent:
Q.sub.abs.about.Re{.epsilon.}/Im{.epsilon.}. Mode-solving
calculations for this type of disk resonances were performed using
two independent methods: numerically, 2D finite-difference
frequency-domain (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.
TABLE-US-00007 TABLE 7 single disk .lamda./r Q.sup.abs Q.sup.rad Q
Re(.epsilon.) = 147.7, m = 2 20.01 (20.00) 10103 (10075) 1988
(1992) 1661 (1663) Re(.epsilon.) = 65.6, m = 3 9.952 (9.950) 10098
(10087) 9078 (9168) 4780 (4802)
The results for two TE-polarized dielectric-disk subwavelength
modes of .lamda./r.gtoreq.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 subwavelength-disk resonant modes. Note that
disk-material loss-tangent Im{.epsilon.}/Re{.epsilon.}=10.sup.-4
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 low-loss-dielectric object values of Q.sub.rad.gtoreq.2000
and Q.sub.abs.about.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 .epsilon.=147.7 has a whispering gallery mode
with m=2, Qrad=13962, and .lamda./r=17.
[0152] The required values of 8, shown in Table 7, might at first
seem unrealistically large. However, not only are there in the
microwave regime (appropriate for approximately meter-range
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 .epsilon.
could signify instead the effective index of other known
subwavelength surface-wave systems, such as surface modes on
surfaces of metallic materials or plasmonic (metal-like,
negative-.epsilon.) materials or metallo-dielectric photonic
crystals or plasmono-dielectric photonic crystals.
[0153] 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 mode-solver
simulations give .kappa. through the frequency splitting
(=2.kappa.) of the normal modes of the combined system, which are
even and odd superpositions of the initial single-disk modes;
analytically, using the expressions for the separation-of-variables
eigenfields E.sub.1,2(r) CMT gives .kappa. through
.kappa.=.omega..sub.1/2d.sup.3r.epsilon..sub.2(r)E.sub.2*(r)E.sub-
.1(r)/.intg.d.sup.3r.epsilon.(r)|E.sub.1(r)|.sup.2 where
.epsilon..sub.j(r) and .epsilon.(r) are the dielectric functions
that describe only the disk j (minus the constant .epsilon..sub.o
background) and the whole space respectively. Then, for medium
distances D/r=10-3 and for non-radiative coupling such that
D<2r.sub.c, where r.sub.c=m.lamda./2.pi. is the radius of the
radiation caustic, the two methods agree very well, and we finally
find, as shown in Table 8, coupling-to-loss ratios in the range
.kappa./.GAMMA..about.1-50. Thus, for the analyzed embodiments, the
achieved figure-of-merit values are large enough to be useful for
typical applications, as discussed below.
TABLE-US-00008 TABLE 8 two disks D/r Q.sup.rad Q = .omega./2.GAMMA.
.omega./2.kappa. .kappa./.GAMMA. Re(.epsilon.) = 147.7, m = 2 3
2478 1989 46.9 (47.5) 42.4 (35.0) .lamda./r .apprxeq. 20 5 2411
1946 298.0 (298.0) 6.5 (5.6) Q.sup.abs.apprxeq. 10093 7 2196 1804
769.7 (770.2) 2.3 (2.2) 10 2017 1681 1714 (1601) 0.98 (1.04)
Re(.epsilon.) = 65.6, m = 3 3 7972 4455 144 (140) 30.9 (34.3)
.lamda./r .apprxeq. 10 5 9240 4824 2242 (2083) 2.2 (2.3) Q.sup.abs
.apprxeq. 10096 7 9187 4810 7485 (7417) 0.64 (0.65)
[0154] 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
self-resonant conducting coils, capacitively-loaded 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.1Q.sub.2)}/Q.sub..kappa. or, if the Q's are limited by
external factors, to increase and/or optimize for
Q.sub..kappa..
[0155] System Sensitivity to Extraneous Objects
[0156] In general, the overall performance of particular embodiment
of the resonance-based wireless energy-transfer 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 non-resonant 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(t) inside the resonant object
1 satisfies, to first order:
d a 1 dt - i ( .omega. 1 - i .GAMMA. 1 ) a 1 + i ( .kappa. 11 - e +
i .GAMMA. 1 - e ) a 1 ( 11 ) ##EQU00037##
where again .omega..sub.1 is the frequency and .GAMMA..sub.1 the
intrinsic (absorption, radiation etc.) loss rate, while
.kappa..sub.11-e is the frequency shift induced onto 1 due to the
presence of e and .GAMMA..sub.1-e is the extrinsic due to e
(absorption inside e, scattering from e etc.) loss rate. The
first-order PT model is valid only for small perturbations.
Nevertheless, the parameters .kappa..sub.11-e, .GAMMA..sub.1-e are
well defined, even outside that regime, if a.sub.1 is taken to be
the amplitude of the exact perturbed mode. Note also that
interference effects between the radiation field of the initial
resonant-object mode and the field scattered off the extraneous
object can for strong scattering (e.g. off metallic objects) result
in total radiation -.GAMMA..sub.1-e 's that are smaller than the
initial radiation .GAMMA..sub.1 (namely .GAMMA..sub.1-e is
negative).
[0157] 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 self-resonant coil, the capacitor
plate spacing of a capacitively-loaded loop or coil, the dimensions
of the inductor of an inductively-loaded rod, the shape of a
dielectric disc, etc.) or changing the position of a non-resonant
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.
[0158] 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.
[0159] 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.
[0160] Unlike the frequency shift, the extrinsic loss can be
detrimental to the functionality of the energy-transfer scheme,
because it is difficult to remedy, so the total loss rate
.GAMMA..sub.1[e]=.GAMMA..sub.1+.GAMMA..sub.1-e (and the
corresponding figure-of-merit .kappa..sub.[e]/ {square root over
(.GAMMA..sub.2[e].GAMMA..sub.2[e])}, where .kappa..sub.[e] the
perturbed coupling rate) should be quantified. In embodiments using
primarily magnetic resonances, the influence of extraneous objects
on the resonances is nearly absent. The reason is that, in the
quasi-static regime of operation (r<<.lamda.) 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
self-capacitance of the coil or the externally loading capacitor),
therefore extraneous non-conducting 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{.mu.}>1 or magnetic loss Im{.mu.}>0). Since almost all
every-day non-conducting materials are non-magnetic 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.
[0161] As noted above, an extremely important implication of this
fact relates to safety considerations for human beings. Humans are
also non-magnetic and can sustain strong magnetic fields without
undergoing any risk. A typical example, where magnetic fields
B.about.1T are safely used on humans, is the Magnetic Resonance
Imaging (MRI) technique for medical testing. In contrast, the
magnetic near-field required in typical embodiments in order to
provide a few Watts of power to devices is only B.about.10.sup.-4T,
which is actually comparable to the magnitude of the Earth's
magnetic field. Since, as explained above, a strong electric
near-field is also not present and the radiation produced from this
non-radiative scheme is minimal, it is reasonable to expect that
our proposed energy-transfer method should be safe for living
organisms.
[0162] One can, for example, estimate the degree to which the
resonant system of a capacitively-loaded conducting-wire 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<<.lamda.) 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 from the center of the
loop (in the limit r<<D) and the angle .theta. with respect
to the loop axis:
u e ( x ) u m ( x ) = o E ( x ) 2 .mu. o H ( x ) 2 = ( 1 + 1 x 2 )
sin 2 .theta. ( 1 x 2 + 1 x 4 ) 4 cos 2 .theta. + ( 1 - 1 x 2 + 1 x
4 ) sin 2 .theta. ; x = 2 .pi. D p .lamda. S p u e ( x ) dS S p u m
( x ) dS = 1 + 1 x 2 1 + 1 x 2 + 3 x 4 ; x = 2 .pi. D p .lamda. , (
12 ) ##EQU00038##
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. 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 (.theta.=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=10r=3 m the ratio of average electric to
average magnetic energy density would be .about.12% and at D=3r=90
cm it would be .about.1%, and for the loop of r=10 cm at a distance
D.sub.p=10r=Im the ratio would be 33% and at D=3r=30 cm it would be
.about.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.fwdarw.1), both are very small, because the fields
have significantly decayed, as capacitively-loaded 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.
[0163] To provide an estimate of the effect of extraneous objects
on the resonance of a capacitively-loaded loop including the
capacitor fringing electric field, we use the perturbation theory
formula, stated earlier,
.GAMMA..sub.1-e.sup.abs=.omega..sub.1/4.intg.d.sup.3rIm{.epsilon..sub.e(r-
)}|E.sub.1(r)|.sup.2/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.times.30 cm.times.1.5
m and permittivity .epsilon.=49+16i (consistent with human muscles)
residing between the loops and almost standing on top of one
capacitor (.about.3 cm away from it) and find
Q.sub.c-h.sup.abs.about.10.sup.5 and for .about.10 cm away
Q.sub.c-h.sup.abs.about.510.sup.5. Thus, for ordinary distances
(.about.1 m) and placements (not immediately on top of the
capacitor) or for most ordinary extraneous objects e of much
smaller loss-tangent, we conclude that it is indeed fair to say
that Q.sub.c-e.sup.abs.fwdarw..infin.. The only perturbation that
is expected to affect these resonances is a close proximity of
large metallic structures.
[0164] Self-resonant coils are more sensitive than
capacitively-loaded 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,
self-resonant coils are simple to make and can withstand much
larger voltages than most lumped capacitors.
[0165] 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/u.sub.m is reduced or minimized at most of the
desired (according to the application) points in the surrounding
space.
[0166] In embodiments using resonances that are not primarily
magnetic, the influence of extraneous objects may be of concern.
For example, for dielectric disks, small, low-index,
low-material-loss or far-away 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 first-order perturbation theory
formulas
.GAMMA..sub.1-e.sup.rad=.omega..sub.1.intg.d.sup.3rRe{.epsilon..sub.e(r)-
}|E.sub.1(r).sup.2/U
and
.GAMMA..sub.1-e.sup.abs=.omega..sub.1/4d.sup.3rIm{.epsilon..sub.e(r)}|E.-
sub.1(r)|.sup.2/U
where U=1/2.intg.d.sup.3r.epsilon.(r)|E.sub.1(r)|.sup.2 is the
total resonant electromagnetic energy of the unperturbed mode. As
one can see, both of these losses depend on the square of the
resonant electric field tails E.sub.1 at the site of the extraneous
object. In contrast, the coupling rate from object 1 to another
resonant object 2 is, as stated earlier,
.kappa.=.omega..sub.1/2.intg.d.sup.3r.sub.2(r)E.sub.2*(r)E.sub.1(r)/.int-
g.d.sup.3r.epsilon.(r)|E.sub.1(r)|.sup.2
and depends linearly on the field tails E.sub.1 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
(.kappa.>>.GAMMA..sub.1-e), at least for small perturbations,
and thus the energy-transfer 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 first-order
perturbation theory approach. For example, we place a dielectric
disk c close to another off-resonance object of large
Re{.epsilon.}, Im{.epsilon.} 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 Ref{.epsilon.}, Im{.epsilon.}
(such as a wall w), as shown in in FIG. 9b. For distances
D.sub.h/w/r=10.sup.-3 between the disk-center 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 high-loss objects. To examine the influence of
large perturbations on an entire energy-transfer system we consider
two resonant disks in the close presence of both a "human" and a
"wall". Comparing Table 8 to the table in FIG. 9c, the numerical
FDFD simulations show that the system performance deteriorates from
.kappa./.GAMMA..sub.c.about.1-50 to
.kappa.[hw]/.GAMMA..sub.c[hw].about.0.5-10 i.e. only by acceptably
small amounts.
[0167] Inductively-loaded conducting rods may also be more
sensitive than capacitively-loaded coils, since they rely on the
electric field to achieve the coupling.
[0168] System Efficiency
[0169] 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 resonance-based
energy-transfer scheme may be determined, when energy is being
drained from the device at rate .GAMMA..sub.work for use into
operational work. The coupled-mode-theory equation for the device
field-amplitude is
da d dt = - i ( .omega. - i .GAMMA. d [ e ] ) a d + i .kappa. [ e ]
a s - .GAMMA. work a d , ( 13 ) ##EQU00039##
where
.GAMMA..sub.d[e]=.GAMMA..sub.d[e].sup.rad+.GAMMA..sub.d[e].sup.abs=-
.GAMMA..sub.d[e].sup.rad+(.GAMMA..sub.d.sup.abs+.GAMMA..sub.d-e.sup.abs)
is the net perturbed-device loss rate, and similarly we define
.GAMMA..sub.s[e] for the perturbed-source. Different temporal
schemes can be used to extract power from the device (e.g.
steady-state continuous-wave 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(t)=A.sub.se.sup.-i.omega.t, so then the field amplitude
inside the device is a.sub.d (t)=A.sub.de.sup.-i.omega.t with
A.sub.d/A.sub.s=i.kappa..sub.[e]/(.GAMMA..sub.d[e]+.GAMMA..sub.work).
The various time-averaged powers of interest are then: the useful
extracted power is P.sub.work=2.GAMMA..sub.work|A.sub.d|.sup.2, the
radiated (including scattered) power is
P.sub.rad=2.GAMMA..sub.s/d.sup.rad|A.sub.s|.sup.2+2.GAMMA..sub.d[e].sup.r-
ad|A.sub.d|.sup.2, the power absorbed at the source/device is
P.sub.s/d=2.GAMMA..sub.s/d.sup.abs|A.sub.s/d|.sup.2, and at the
extraneous objects
P=2.GAMMA..sub.s-e.sup.abs|A.sub.s|.sup.2+2.GAMMA..sub.d-e.sup.abs|A.sub.-
d|.sup.2. From energy conservation, the total time-averaged power
entering the system is
P.sub.total=P.sub.work+P.sub.rad+P.sub.s+P.sub.d+P.sub.e. 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 power-balance calculations. The working
efficiency is then:
.eta. work .ident. P work P total = 1 1 + .GAMMA. d [ e ] .GAMMA.
work . [ 1 + 1 fom [ e ] 2 ( 1 + .GAMMA. work .GAMMA. d [ e ] ) 2 ]
, ( 14 ) ##EQU00040##
where fom.sub.[e]=.kappa..sub.[e]/ {square root over
(.GAMMA..sub.s[e].GAMMA..sub.d[e])} is the distance-dependent
figure-of-merit of the perturbed resonant energy-exchange system.
To derive Eq. (14), we have assumed that the rate
.GAMMA..sub.supply, at which the power supply is feeding energy to
the resonant source, is
.GAMMA..sub.supply=.GAMMA..sub.s[e]+.kappa..sup.2/(.GAMMA..sub.d[e]+.GAMM-
A..sub.work), such that there are zero reflections of the fed power
P.sub.total back into the power supply.
Example: Capacitively-Loaded Conducting Loops
[0170] 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 capacitively-loaded
conducting loops, one can consider the following circuit-model of
the system, where the inductances L.sub.s, L.sub.d represent the
source and device loops respectively, R.sub.s, R.sub.d their
respective losses, and C.sub.s, C.sub.d are the required
corresponding capacitances to achieve for both resonance at
frequency .omega.. A voltage generator V.sub.g is considered to be
connected to the source and a work (load) resistance R.sub.w to the
device. The mutual inductance is denoted by M.
[0171] Then from the source circuit at resonance
(.omega.L.sub.d=1/.omega.C.sub.d):
V.sub.g=I.sub.sR.sub.s-j.omega.MI.sub.d1/2V.sub.g*I.sub.s=1/2|I.sub.s.su-
p.2R.sub.s+1/2j.omega.MI.sub.d*I.sub.s,
and from the device circuit at resonance
(.omega.L.sub.d=1/.omega.C.sub.d):
0=I.sub.d(R.sub.d+R.sub.w)-j.omega.MI.sub.sj.omega.MI.sub.s=I.sub.d(R.su-
b.d+R.sub.w)
So by substituting the second to the first:
1/2V.sub.gI.sub.s=1/2|I.sub.s|.sup.2R.sub.s+1/2|I.sub.d|.sup.2(R.sub.d+R-
.sub.w).
Now we take the real part (time-averaged powers) to find the
efficiency:
P g .ident. Re { 1 2 V g * I s } = P s + P d + P w .eta. work
.ident. P w P tot = R w I s I d 2 R s + R d R w . Namely , .eta.
work = R w ( R d + R w ) 2 ( .omega. M ) 2 R s + R d + R w ,
##EQU00041##
which with .GAMMA..sub.work=R.sub.w/2L.sub.d,
.GAMMA..sub.d=R.sub.d/2L.sub.d, .GAMMA..sub.s=R.sub.s/2L.sub.s, and
.kappa.=.omega.M/2 {square root over (L.sub.sL.sub.d)}, becomes the
general Eq. (14). [End of Example]
[0172] From Eq. (14) one can find that the efficiency is optimized
in terms of the chosen work-drainage rate, when this is chosen to
be
.GAMMA..sub.work/.GAMMA..sub.d[e]=.GAMMA..sub.supply/.GAMMA..sub.s[e]=
{square root over (1+fom.sub.[e].sup.2)}>1. Then, .eta..sub.work
is a function of the fom.sub.[e] parameter only as shown in FIG. 11
with a solid black line. One can see that the efficiency of the
system is .eta.>17% for fom.sub.[e]>1, large enough for
practical applications. Thus, the efficiency can be further
increased towards 100% by optimizing fom.sub.e, as described above.
The ratio of conversion into radiation loss depends also on the
other system parameters, and is plotted in FIG. 11 for the
conducting loops with values for their parameters within the ranges
determined earlier.
[0173] For example, consider the capacitively-loaded coil
embodiments described in Table 4, with coupling distance D/r=7, a
"human" extraneous object at distance D.sub.h from the source, and
that P.sub.work=10 W must be delivered to the load. Then, we have
(based on FIG. 11)
Q.sub.s[h].sup.rad=Q.sub.s[h].sup.rad.about.10.sup.4,
Q.sub.s.sup.abs=Q.sub.d.sup.abs.about.10.sup.3,
Q.sub..kappa..about.500, and Q.sub.d-h.sup.abs.fwdarw..infin.,
Q.sub.s-h.sup.abs.about.10.sup.5 at D.sub.h.about.3 cm and
Q.sub.s-h.sup.abs.about.510.sup.5 at D.sub.h.about.10 cm. Therefore
fom.sub.[h].about.2, so we find .eta.=38%, P.sub.rad.about.1.5 W,
P.sub.s.apprxeq.11 W, P.sub.d.apprxeq.4 W, and most importantly
.eta..sub.h.apprxeq.0.4%, P.sub.h=0.1 W at D.sub.h.about.3 cm and
.eta..sub.h.apprxeq.0.1%, P.sub.h=0.02 W at D.sub.h.about.10
cm.
[0174] Overall System Performance
[0175] 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 cell-phone, the device
resonant object cannot have dimensions larger that those of the
laptop or cell-phone respectively. In particular, for a system of
two loops of specified dimensions, in terms of loop radii r.sub.s,d
and wire radii a.sub.s,d, the independent parameters left to adjust
for the system optimization are: the number of turns N.sub.s,d, the
frequency f the work-extraction rate (load resistance)
.GAMMA..sub.work and the power-supply feeding rate
.GAMMA..sub.supply.
[0176] In general, in various embodiments, the primary dependent
variable that one wants to increase or optimize is the overall
efficiency .eta.. However, other important variables need to be
taken into consideration upon system design. For example, in
embodiments featuring capacitively-loaded coils, the design may be
constrained by, for example, the currents flowing inside the wires
I.sub.s,d and the voltages across the capacitors V.sub.s,d. These
limitations can be important because for .about.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=.omega.L.sub.d/(R.sub.d+R.sub.w) 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 should be minimized for safety concerns, even
though, in general, for a magnetic, non-radiative scheme they are
already typically small.
[0177] 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 work-drainage rate for some particular
value of fom.sub.[e] through .GAMMA..sub.work/.GAMMA..sub.d[e]=
{square root over (1+wpfom.sub.[e].sup.2)}. Then, in some
embodiments, values which impact the choice of this rate are:
.GAMMA..sub.work/.GAMMA..sub.d[e]=1.revreaction.wp=0 to minimize
the required energy stored in the source (and therefore I.sub.s and
V.sub.s), .GAMMA..sub.work/.GAMMA..sub.d[e]= {square root over
(1+fom.sub.[e].sup.2)}>1.revreaction.wp=1 to increase the
efficiency, as seen earlier, or
.GAMMA..sub.work/.GAMMA..sub.d[e]>>1.revreaction.wp>>1
to decrease the required energy stored in the device (and therefore
I.sub.d and V.sub.d) and to decrease or minimize
Q.sub.tot=.omega.L.sub.d/(R.sub.d+R.sub.w)=.omega./[2(.GAMMA..sub.d+.GAMM-
A..sub.work)]. Similar is the impact of the choice of the power
supply feeding rate .GAMMA..sub.supply, with the roles of the
source and the device reversed.
[0178] Increasing N.sub.s and N.sub.d increases
.kappa./.GAMMA..sub.s.GAMMA..sub.d and thus efficiency
significantly, as seen before, and also decreases the currents
I.sub.s and I.sub.d, because the inductance of the loops increases,
and thus the energy U.sub.s,d=1/2L.sub.s,d|I.sub.s,d|.sup.2
required for given output power P.sub.work can be achieved with
smaller currents. However, increasing N.sub.d increases Q.sub.tot,
P.sub.rad,d and the voltage across the device capacitance V.sub.d,
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 implies reduced required
capacitance C.sub.d, which could be achieved in principle, for a
capacitively-loaded device coil by increasing the spacing of the
device capacitor plates d.sub.d and for a self-resonant coil by
increasing through h.sub.d the spacing of adjacent turns, resulting
in an electric field (.apprxeq.V.sub.d/d.sub.d for the former case)
that actually decreases with N.sub.d; however, one cannot in
reality increase d.sub.d or h.sub.d 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 and V.sub.s upon increasing N.sub.s. As a conclusion,
the number of turns N.sub.s and N.sub.d have to be chosen the
largest possible (for efficiency) that allow for reasonable
voltages, fringing electric fields and physical sizes.
[0179] With respect to frequency, again, there is an optimal one
for efficiency, and Q.sub.tot 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.
[0180] 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=25 cm, r.sub.d=15 cm, h.sub.s=h.sub.d=0, a.sub.s=a.sub.d=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, given some
choice for wp and N.sub.s. 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 and N.sub.d 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=2, N.sub.d=6, f=10 MHz and
wp=10, which gives the following performance characteristics:
.eta..sub.work=20.6%, Q.sub.tot=1264, I.sub.s=70.2 A, I.sub.d=1.4
A, V.sub.s=2.55 kV, V.sub.d=2.30 kV, P.sub.rad,s=0.15 W,
P.sub.rad,d=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, N.sub.d, still
they give a good estimate of the scalings and the orders of
magnitude.
[0181] 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 and a.sub.s 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.
[0182] Experimental Results
[0183] An experimental realization of an embodiment of the above
described scheme for wireless energy transfer consists of two
self-resonant 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 ("light-bulb"). The
various .kappa.'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.
[0184] 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=10.56.+-.0.3 MHz, which is about 5% off from the measured
resonance at around 9.90 MHz.
[0185] The theoretical Q for the loops is estimated to be
.about.2500 (assuming perfect copper of resistivity
p=1/.sigma.=1.7.times.10.sup.-8 .OMEGA.m) 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 (.about.20 .mu.m) at this frequency. We have therefore used
the experimentally observed Q (and
.GAMMA..sub.1=.GAMMA..sub.2=.GAMMA.=.omega./(2Q) derived from it)
in all subsequent computations.
[0186] The coupling coefficient K can be found experimentally by
placing the two self-resonant coils (fine-tuned, 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
coupled-mode theory, the splitting in the transmission spectrum
should be .DELTA..omega.=2 {square root over
(.kappa..sup.2-.GAMMA..sup.2)}. 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.
[0187] FIG. 16 shows a comparison of experimental and theoretical
values for the parameter .kappa./.GAMMA. as a function of the
separation between the two coils. The theory values are obtained by
using the theoretically obtained K and the experimentally measured
.GAMMA.. The shaded area represents the spread in the theoretical
.kappa./.GAMMA. due to the .about.5% uncertainty in Q.
[0188] As noted above, the maximum theoretical efficiency depends
only on the parameter .kappa./ {square root over
(.GAMMA..sub.1.GAMMA..sub.2)}=.kappa./.GAMMA., plotted as a
function of distance in FIG. 17. The coupling to loss ratio
.kappa./.GAMMA. is greater than 1 even for D=2.4 m (eight times the
radius of the coils), thus the system is in the strongly-coupled
regime throughout the entire range of distances probed.
[0189] 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 light-bulb, 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 light-bulb and the device coil, the parameter
.GAMMA..sub.work/.GAMMA. was adjusted so that it matched its
optimal value, given theoretically by {square root over
(1+.kappa..sup.2/(.GAMMA..sub.1.GAMMA..sub.2))}. Because of its
inductive nature, the loop connected to the light-bulb added a
small reactive component to .GAMMA..sub.work 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 light-bulb at the load was at its full nominal
brightness.
[0190] 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 mid-point of each of the self-resonant
coils with a current-probe (which was not found to lower the Q of
the coils noticeably.) This gave a measurement of the current
parameters I.sub.1 and I.sub.2 defined above. The power dissipated
in each coil was then computed from
P.sub.1,2=.GAMMA.L|I.sub.1,2|.sup.2, and the efficiency was
directly obtained from
.eta.=P.sub.work/(P.sub.1+P.sub.2+P.sub.work). To ensure that the
experimental setup was well described by a two-object coupled-mode
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).
[0191] Using this embodiment, we were able to transfer significant
amounts of power using this setup, fully lighting up a 60 W
light-bulb 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
wall-to-load efficiency of .about.15%, which is reasonable given
the expected .about.40% efficiency for the wireless power transfer
at that distance and the low efficiency of the driving circuit.
[0192] 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
.DELTA.f/f.sub.0 of a few times the inverse loaded Q, the induced
current in the device coil was indistinguishable from noise.
[0193] 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.
[0194] 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 long-term use. The power radiated for these parameters
was .about.5 W, which is roughly an order of magnitude higher than
cell phones but could be drastically reduced, as discussed
below.
[0195] 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.
[0196] 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.
[0197] We believe that the efficiency of the scheme and the
distances covered could be appreciably improved by silver-plating
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.
[0198] Applications
[0199] In conclusion, we have described several embodiments of a
resonance-based scheme for wireless non-radiative energy transfer.
Although our consideration has been for a static geometry (namely
.kappa. and .GAMMA..sub.e were independent of time), all the
results can be applied directly for the dynamic geometries of
mobile objects, since the energy-transfer time .kappa..sup.-1
(.about.1 .mu.s-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 modern applications.
[0200] 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 source-object could be an elongated "pipe" running
above the highway, or along the ceiling.
[0201] 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.
[0202] In the microscopic world, where much smaller wavelengths
would be used and smaller powers are needed, one could use it to
implement optical inter-connects for CMOS electronics, or to
transfer energy to autonomous nano-objects (e.g. MEMS or
nano-robots) 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 condensed-matter
object.
[0203] In some embodiments, the techniques described above can
provide non-radiative wireless transfer of information using the
localized near fields of resonant object. Such schemes provide
increased security because no information is radiated into the
far-field, and are well suited for mid-range communication of
highly sensitive information.
[0204] 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.
* * * * *