Optimization of Wireless Power Transfer Systems Enhanced by Passive Elements and Metasurfaces

2.6 MB·

by  Hans-Dieter Lang

IEEETRANSACTIONSONANTENNASANDPROPAGATION 1
Optimization of Wireless Power Transfer Systems
Enhanced by Passive Elements and Metasurfaces
Hans-Dieter Lang, Student Member, IEEE, and Costas D. Sarris, Senior Member, IEEE
Abstract—This paper presents a rigorous optimization tech- Multiple passive elements can be used cooperatively; for
nique for wireless power transfer (WPT) systems enhanced by example to form near-field guiding structures [11]–[15] or
7 passive elements, ranging from simple reflectors and intermedi- provide more general functions associated with metasurfaces
1 ate relays all the way to general electromagnetic guiding and or metamaterials [16]–[18].
0 focusing structures, such as metasurfaces and metamaterials. At
As will be shown, the optimization of such systems is
2 its core is a convex semidefinite relaxation formulation of the
otherwise nonconvex optimization problem, of which tightness not a trivial task; the standard formulations of the required
r
a andoptimalitycanbeconfirmedbyasimpletestofitssolutions. passivity and power constraints are nonconvex. This renders
M Theresultingmethodisrigorous,versatile,andgeneral—itdoes the whole problem unsolvable in general, particularly for
not rely on any assumptions. As shown in various examples, it
a large number of unknowns (e.g. the number of passive
is able to efficiently and reliably optimize such WPT systems in
1 elements). Here, however, a convex relaxation formulation is
order to find their physical limitations on performance, optimal

operating parameters and inspect their working principles, even derived, which essentially constitutes a general and rigorous
]
C for a large number of active transmitters and passive elements. work-aroundtothisproblem.Furthermore,asimpletestofthe
O Index Terms—Convex Optimization, Multiple Transmitters, solution confirms tightness (meaning exact representation of
Passive Couplers, Power Transfer Efficiency, Semidefinite Pro- the original problem) and hence, that the true global optimum
.
h gramming, Tight Relaxation, Wireless Power Transfer. has been found.
t
a The resulting powerful and rigorous optimization method
m I. INTRODUCTION generalizes the previously presented optimization technique
[ WIRELESS power transfer (WPT) systems have been forWPTsystemswithmultipleactivetransmitters[19].Hence,
2 investigated in various forms, contexts and for various it can be used to investigate the maximum performance of a
particularsystemwithmultipleactivetransmittersandpassive
v applications [1]–[4]. A large variety of unique solutions for
elements and to obtain the corresponding optimal operating
4 powering and charging devices in an untethered fashion has
9 parameters(i.e.currentsandloadingreactances).Furthermore,
been proposed; ranging from applications such as charging
6 using an outer loop optimization, the optimum load resistance
mobile devices [5] to electric vehicles [6]–[8] and including
7 can also be found, leading to the maximum achievable PTE
0 charging,operatingaswellascommunicatingwithbiomedical
of the system; the absolute limit on the performance.
. implants [9].
1 The outline of this paper is as follows: After a short
However,amongotherconstraints,theperformance—most
0
introduction, the optimization problem of passively-enhanced
7 commonly measured by the power transfer efficiency (PTE),
WPTsystemsispresented.Theconvexsemidefiniterelaxation
1 the efficiency at which the power can be transferred from the
: transmitter(s) to the receiver—considerably limits the range is then derived and a simple test for tightness is given. In the
v
results section, three numerical examples prove the validity
i of such applications. Usually, PTEs high enough for practical
X and versatility of the method: First, the case where additional
applications are confined to distances between transmitter(s)
r and receiver(s) in the order of fractions of the wavelength at passively excited transmitters are added behind the active
a
transmitterisconsidered.Second,optimalrelayconfigurations
the operating frequency as well as to dimensions comparable
are treated, useful for both high-efficiency range extension as
to those of the transmitter and receiver coils [5].
well as for misalignment mitigation. Lastly, the enhancement
Various attempts to mitigate this problem and increase the
by general passive metasurfaces is discussed; proving the
PTE or extend the range of high PTE have been proposed.
method’s capability to handle a large number of passive
A popular method to enhance PTE involves introducing pas-
elements.Attheend,finalremarksandconclusionsaregiven.
sive elements in the proximity of the transmitter(s) and the
Remarks on the notation: Thin italic letters represent scalar
receiver[10],actingasrelayelementsorasparasiticelements
variables,boldsmalllettersrefertovectors,boldcapitalletters
that can enlarge the effective aperture of the transmitter.
arematrices;vT standsforthetransposeofthevectorv,while
Manuscript received May xx, 2016; revised Month xx, 2016; accepted vH standsforitsHermitian(conjugatetranspose).Thesymbol
Monthxx,2016.Dateofpublication:Monthzz,2016.Currentversion:Later ( ) is used to denominate positive (semi-) definiteness
00 (cid:31)of m(cid:23)atrices, respectively. A star ()? marks the optimized
The authors are with the Electromagnetics Group of the Department of ·
ElectricalandComputerEngineering,UniversityofToronto,ON,M5S3G4, arguments leading to the optimal solution, while a bar (a¯)
Canada,(e-mail:[email protected],[email protected]). is used to denote the complex conjugate. Lastly, ’program’ is
Color versions of one or more of the figures in this paper are available
used as a synonym for ’optimization problem’, as is common
onlineathttp://ieeexplore.ieee.org.
DigitalObjectIdentifierXX.YYYY/TAP.2016.ZZZZZZZ in the context of mathematical optimization [20].
IEEETRANSACTIONSONANTENNASANDPROPAGATION 2
II. PRELIMINARIES complex (leading to non-zero real parts in the off-diagonal
entries of Z), due to retardation effects, when the electrical
A. The MISO WPT System Model with Passive Elements
distancebetweenthetransmitter(s)and/orreceiverarenotvery
Fig. 1 depicts the general form of wireless power transfer
small.
(WPT) systems under consideration, incorporating multiple
For physical reasons, impedance matrices of passive recip-
active transmitters, multiple passive elements and a single
rocal circuits have to be positive-real [21]: The impedance
receiver. Systems with multiple transmitters will be referred
to as MISO (multiple-input, single-output) WPT systems, in matrix is symmetric Z = ZT and all Z0,Z0t,Z0a,Z0p (cid:31) 0 and
z > 0 (positive-definite), meaning positive quadratic forms,
contrasttoSISO(single-inputsingle-output)systemswithonly r0
for example:
a single transmitter.
iHZ0i>0 i CN (2)
∀ ∈
LoadedWPTS: Zˆ=Z+jX+RL Positive semidefiniteness ((cid:23) 0) implies nonnegative (≥ 0)
quadratic forms. For negative (semi-) definiteness the oppo-
UnloadedWPTS: Z site signs and directions apply. In the context of impedance
C
i1 C11 TX RX Cr matrices, the mathematical property of positive-definiteness
v1 Zt zr represents the fact that there is always some amount of loss
i
r in the system.
i2 ztr vr=0 The process of adding reactances to each of the transmitter
v2 C2 RL awnidllrbeecerievfeerrrneoddetosaasswloealldainsgadodfinthgeaWrePsiTstisvyestleomad,awshreerceeiver
X
R
veTX C3 CN−1 passive is the resulting loadedZˆi=mpZed+anjcXe m+aRtriLx, marked by a hat.(I3n)
cti
a full detail, the voltages and currents v, i of the entire WPT
i i
3 N 1
passiveelements v =0 v −=0 system are related according to
3 N 1
−
v X 0 i
a a a
Fig. 1. Loop-based MISO WPT system model with multiple active trans-
v =0 = Z+j X + 0 i
mittersandpassiveelements:Corestructurewithunloadedimpedancematrix  p    p     p
Z, reactive components xn = −(ωCn)−1 (in case of capacitors), resistive vr =0 xr RL ir
receiverloadRL,andvoltagesourcesvn.Inthisexample,subscripts1and         
2 correspond to active transmitters (i.e. connected to a source), while 3 to v X R i
L
N 1standforthepassiveelements.
− | {z } | {z } | {z } |{z}(4)
Let the (unloaded) impedance matrix Z CN×N of such where the subscripts t, a, p, and r have the same meanings as
∈
systems be partitioned according to explained for (1). The real-valued diagonal matrix X contains
the load reactances, where positive and negative values refer
Za Zap zar to inductive and capacitive loading, respectively. They are
    grouped into the two diagonal matrices corresponding to the
Zt ztr activetransmittersandpassiveelementsaswellasascalarfor
Z= = ZTap Zp zpr . (1) the receiver. The matrix RL is zero everywhere but at the last
   
    diagonal entry, corresponding to the receiver, where the load
 zTtr zr   zTar zTpr zr  resistance RL >0 is located.
   
    It is important to note that, since the load resistance RL
The subscripts t and r refer to the transmitter and receiver
is actually part of the impedance matrix, KVL states that
parts, respectively, whereas the subscripts a and p refer to the
the corresponding receiver voltage is zero, i.e. v = 0 as
r
active and passive transmitters. Finally, the combinations tr,
explicated in (4) and depicted in Fig. 1. The same is true
ap, ar and pr stand for the submatrices and vectors coupling
for the voltages of the passive couplers, v =0.
p
two of these groups together.
Let all the nodes be numbered by n = 1,...,N, where
B. The Objective: Power Transfer Efficiency (PTE)
N = A + P is the total number of nodes, including A
active transmitters, n (note that Za CA×A) and P The central figure of merit when optimizing WPT systems
passivenodes,n —∈noAtethatthisinclude∈sboththepassive is the power transfer efficiency1 (PTE) [1]–[5], [7], [9]: the
elements and the∈rePceiver. ratio of the power PL transferred to the load RL, to the total
The diagonals of Z, Zt, Za, and Zp as well as zr refer to transmit power provided by the source Pt
the loss resistances and self-reactances of the transmitters and P P R
L L L
η = = = . (5)
receiver, respectively. Typically, all Z0,0nn, zr00 > 0 (inductive) Pt Pl+PL Rl+RL
when considering systems of magnet·ically coupled loops or
coils; however, this is not a requirement. Likewise, all off- 1It should be noted that the so defined PTE refers only to the electro-
magnetic power transfer efficiency; i.e. the impacts of impedance matching,
diagonal entries of Z are the coupling impedances due to
rectificationandgenerationontothetotalefficiencyofanentirewirelesspower
the mutual inductances jωMn,m. Generally each Mn,m is transfersystemarenotconsidered,here.
IEEETRANSACTIONSONANTENNASANDPROPAGATION 3
P isthetotalpowerabsorbedbythesystem,duetodissipation Gathering all these constraints, the following program is
l
and radiation, modeled by the loss resistance R . obtained:
l
The goal of this paper is to find a mathematical method
to determine the absolute performance limits of such WPT 1
systems to wirelessly transfer power most efficiently via Pl,min =miin 2iHZ0i (power loss)
multiple active transmitters and passive elements to a single 1
receiver. Hence, the aim is to maximize the PTE, obtained s.t. iHTˆni 0 n (active nodes)
2 ≥ ∈A
according to its definition (5) as the biquadratic form (8)
1
iHTˆ i=0 n (passive nodes)
1iHR i iHR i 2 n ∈P
η = 2 L = L . (6)
12(iHv)0 iH(Z0+RL)i 1iHRLi=1 (transferred power)
2
by finding the corresponding optimal voltages v and currents
i (real and imaginary parts) as well as loading elements
x = diag(X) = x ,x ,x and R . It is most convenient This a nonconvex quadratically constrained quadratic pro-
a p r L
to formulate the PTE in terms of the currents i. The voltages gram(QCQP)[20],becausesome(inthiscaseall)constraints
v can be obtaine(cid:2)d from the(cid:3)currents using the fully loaded are quadratic and nonconvex. Such problems are well-known
to be notoriously difficult to solve (as they belong to the class
system (4). However, both are only meaningful solutions as
long as v =0 and v =0. ofNP-hardproblems).Mostimportantly,therearenoavailable
r p
algorithms which are able to solve these types of problems in
general, for an arbitrary number of unknowns, in polynomial
III. OPTIMIZATION time.
A. Port Impedance Matrices (PIMs)
The total power inserted into the system P in the denom-
t
inator of (5) can be separated into the contributions of each
C. Tight Semidefinite Relaxation
transmitter as follows [19]:
1 1
P =P +P = (iHv) = iHZˆ i The following derivation follows in essence the most com-
t l L 0 0
2 2 mon standard semidefinite relaxation techniques layed out in
1
= P = iHTˆ i (7) detail in any reference on the matter, see e.g. [22], and will
t,n n
2 therefore only be given in brief form.
n n
X X
where Tˆ are the (loaded) port impedance matrices (PIMs), 1) Nonconvex QCQP: The phases of the currents are only
n
and n denotes the port. Note that T =Tˆ , for all n except defined relative to each other. Thus, the phase of one the
n n
n = N (receiver), where Tˆ = T +R ; i.e. loading only currents can be chosen freely, without any loss in generality;
N N L
affects the Nth PIM. any other absolute phase solution could then be obtained by
WhilethePIMssumuptothetotallossresistancematrices, renormalization. The most natural choice is a purely real
di.eefi.niten),Tˆthne=PIMZˆs0 t(cid:31)hem0s(erlevaels-vaarleuecdo,mspylmexm-veatlruiceda,nHderpmositiitaivne, rpercoebilveemr csiuzrerecnatn, ib.ee.rierdu=cedi0rtoanMd =i0r02=N 0.1T.hTuhse, trheemaaicntiunagl
−
andPindefinite. Furthermore, each PIM is singular, as it has unknowns are the currents (separated into their real and
exactlyonepositive,onenegativeandN 2zeroeigenvalues. imaginary parts)
−
These eigenvalues and eigenvectors can be obtained analyti-
cally, as shown in Appendix A. i0a
i0t i0p
B. Quadratically-Constrained Quadratic Program (QCQP) c=i0r= i0r ∈RM . (9)
The goal is to minimize the total power required for unit i0t0 i0a0
transferred power 21iHRLi= 1 in terms of complex currents   i0p0
i under the following additional constraints:  
All power fluxes at ports of active nodes (transmitters) Usingthecyclicpropertyofthetrace,cTQ c=tr(cTQ c)=
• be nonnegative, i.e. 12iHTˆni≥0, ∀n∈A (the set of all tr(QiccT), the objective and all constraintsi can be writtien in
A active nodes), ensuring power is fed into the system linear terms of the (quadratic) current matrix C=ccT 0:
(cid:23)
via each active transmitter port, rather than drained from
it [19].
P =min tr(Q C)
All power fluxes at ports of passive nodes (passive l,min c 0
• elements and the receiver) be zero, i.e. 1iHTˆ i = 0, s.t. tr(Q C) 0 n
2 n n ≥ ∈A
n (the set of all P passive nodes). This represents
∀the∈faPct that there is neither power fed into nor drained tr(QnC)=0 n∈P (10)
fromthesystem.Notethatthisstillallowsforpowerloss tr(RC)=1
in that passive element. C=ccT
IEEETRANSACTIONSONANTENNASANDPROPAGATION 4
where λ 0, in order to only punish tr(Q C)<0, n ,
• n ≥ n ∀ ∈A
Tˆ0 Tˆ00 • νn ∈R, penalizing tr(QnC)6=0, ∀n∈P, and finally
Q = n − n n=1,...,N (11) σ R, responsible for punishing tr(RC)=1.
n "Tˆ0n0 Tˆ0n #M M To•shor∈ten the notation, the first two groups of6dual variables
×
Q = Z0+RL −Z0n0 = N Q (12) areTrheeferprreidmtaolapsrvoebcletomrs(λi.e=. [tλhne]oarnidgiνna=l p[νronb],lermespe(1c4ti)v)elyis.
0 (cid:20) Z0n0 Z0+RL(cid:21)M M n=1 n obtained from the Lagrangian by minimizing the objective
× X
R 0 while an inner maximization ensures that all constraints are
R= L =Diag 0N 1,R ,0N 1 (13)
0 R − L − satisfied:
(cid:20) L(cid:21)M×M (cid:2) (cid:3) Pprimal =min maxL. (16)
The subscripts of the matrices refer to the size of the leading l,min C 0 λ 0
(cid:23) ν(cid:23),σ
principal submatrices thereof; the last row and column are
dropped, because they would refer to the imaginary part of Note that, if one or more of the constraints were not satisfied,
the receiver current, which is chosen to be zero at all times, theLagrangianwouldbeunboundedanddivergeto+ .Thus,
∞
as discussed. the primal problem is (16) is really identical to (14).
Notethattheprograms(8)and(10)aremathematicallyfully Theso-calleddualproblemisobtainedbyinterchangingthe
equivalent.Thenonconvexitiesofthequadraticinequalityand order of the maximization and minimization:
equality constraints has been isolated in the last (nonconvex)
Pdual =max minL=max σ : Q 0 . (17)
equality constraint, requiring the matrix C to be the outer l,min λ 0 C 0 λ 0 { (cid:23) }
product of the actual vector of unknowns c. All the other ν(cid:23),σ (cid:23) ν(cid:23),σ
constraints are affine (or even linear) in C. Notethatthedualprogram(17)takesthesameformasin[19].
2) Semidefinite Relaxation: As for any convex relaxation, However, some constraints on the dual variables λ 0
n
≥
the idea is to exchange the nonconvex constraint by a con- (active transmitters) within Q have been relaxed to νn R
∈
straint that achieves almost the same—while at the same (passive elements), thereby broadening the solution space and
time being convex. The constraint C = ccT is equivalent allowing for larger σ, meaning lower optimal PTE η.
to the so-called rank-1 condition rankC = 1, which also As long as the primal problem satisfies certain conditions
implies C 0. To obtain a convex semidefinite relaxation, [20] (which are satisfied in this case, since the problem is
(cid:23)
therank-1conditionisremoved,whilestillrequiringpositive- continuous and convex), the dual optimum (17) is equivalent
semidefiniteness2 . totheprimaloptimum(16).Therefore,inthiscase,theoptimal
Thus,thefollowingsemidefiniteprogram(SDP)isobtained: solutionsofthetwoproblemsarethesame,Pprimal =Pdual.
l,min l,min
Prelax =min tr(Q C) 4) Optimalityconditions: TheKarush-Kuhn-Tucker(KKT)
l,min C 0 conditions of optimality [20] for the two problems are:
s.t. tr(Q C) 0 n
n ≥ ∈A C? 0 (primal feasibility) (18a)
tr(Q C)=0 n (14) (cid:23)
n ∈P Q? 0 (dual feasibility 1) (18b)
(cid:23)
tr(RC)=1 λ? 0 (dual feasibility 2) (18c)
(cid:23)
C 0. tr(RC?)=1 (primal equality 1) (18d)
(cid:23)
This is a convex program which can be solved reliably and tr(Q C?) 0 n (primal inequalities) (18e)
n ≥ ∈A
efficiently using dedicated algorithms. In Matlab, it can be tr(Q C?)=0 n (primal equalities) (18f)
implemented comfortably using CVX [23], [24] and solved n ∈P
λ tr(Q C?)=0 n (compl. slackness 1) (18g)
for example using the standard solver SDPT3 [25], [26]. n n ∈A
3) Duality: The Lagrangian, the weighted (penalized) sum tr(Q?C?)=0 (complementary slackness 2) (18h)
oftheobjectiveandconstraints[20]forthisSDP(14)isgiven
The last condition, the complementary slackness, boils down
by:
to Q?C? =0, since C?,Q? 0. In the case of rankC? =1
this also means that rankQ?(cid:23)=M 1, or in other words that
L=tr "Q0− λnQn− νnQn−σR#C!+σ Q? has exactly one zero eigenvalue−and c? is the correspond-
nX∈A nX∈P ing eigenvector, the sought-after solution to the problem
(15a)
5) Tightness: Since the constraints of the actual prob-
=tr [P σR]C +σ (15b) lem (10) have been loosened (essentially simply by removing
−
one of them), the optimal value of (14) is generally only a
=tr(cid:0)QC σ (cid:1) (15c) lower bound on the actual optimum, i.e. Prelax P .
− l,min ≤ l,min
Therefore, by application of (5), the solution is only an upper
where the(cid:0) dua(cid:1)l variables (penalty factors) are:
bound on the maximum possible PTE, i.e. ηrelax η .
max ≥ max
2AnotherinterpretationofC=ccT isthatC ccT =0,equivalentto Equality is only achieved in the case of (14) being a tight
thetwosemidefiniteconstraintsC ccT 0and−C ccT 0.Then,the (meaningexact)relaxationof(8)and(10).Thiswouldrequire
− (cid:23) − (cid:22)
relaxation takes place by removing the second constraint and using the first
alone,byrequiringthematrixofwhichC ccT istheSchurcomplement, that the removed rank constraint is naturally satisfied by the
tobepositive-semidefinite[19]. − rest of the problem.
IEEETRANSACTIONSONANTENNASANDPROPAGATION 5
To test tightness, either the eigenvalues of C? or Q? could configurationsprovidedfromtheframeworkpresentedin[4]is
be investigated, or, numerically more efficient, the (normal- usuallyveryclosetothetrueoptimumandasimpleandcom-
ized) tightness error, defined as putationallyefficientalternative.Thecalculationonlyrequires
C? c?(c?)T the minimum-loss output impedance zo and (the square of)
(cid:15)= − (19) the mutual coupling quality factor U, both directly obtainable
(c?)Tc?
(cid:13) (cid:13) from the unloaded impedance matrix:
(cid:13) (cid:13)
can be used. Whenever (cid:15) is small (i.e. within the bounds of
typical numerical approximation errors of such algorithms), zo =zr−ztrZ0t−1z0tr (23)
tohpetimseummidseofiluntiitoenrce?laxtoattihoenniosnctiognhvtexanQdCpQrPov(i1d0e)s. an exact U2 = zHtrZz0t−1ztr (24)
o0
Numerical experiments showed that this seems always the
case for practical setups (typically down to (cid:15) 10−14...−10). RL? =zo0 1+U2 (25)
Thus, C? has exactly one nonzero (and in≈deed positive) wherethepartitionsareaccoprdingto(1).Asdiscussedin[27],
eigenvalue and c? is its corresponding eigenvector; the global
thesearethephysicallymeaningfulgeneralizationsofthewell-
optimum to (10).
known SISO WPT systems [3].
D. Optimal Load Reactances and Voltages
IV. APPLICATIONEXAMPLES
The reactive loads in all passive nodes (including the
It is well understood that the region of high PTE is
receiver) are obtained from solving the corresponding homo-
usually limited to electrically very short distances between
geneous linear equations v = 0 and v = 0. Physically, this
p r
transmitter(s) and receiver [5]. Several approaches have been
means that in each of these loops, the voltages across these
proposedinordertoextendtheelectricaldiameterofthehigh-
reactiveelementsarethesamebutwithoppositesignasacross
PTE region, leading to the so-called “mid-range” [9] wireless
everything else (the unloaded WPT system), in order to net in
power transfer systems. The most straightforward ideas are
zero voltages overall.
shielding, guiding (near-field beamforming) and focusing; all
To this end, the unloaded voltages are calculated according
of which will be addressed briefly in the following via the
to
v? =Zi? . (20) three examples depicted in Fig. 2.
unloaded
Then, the optimal load reactances for the passive nodes (n
∈
P) can be obtained using z z z
x?n =−(cid:18)vu?nloia?nded,n(cid:19)00 . (21) xrRL RX RxRrXL Praeslasyivse xrRL MeRtXasurface
Itcaneasilybeverifiedthatthezerorealpowerfluxconstraint TX y
at port n, P = 1(i?)HTˆ i? = 1(v? i?) = 0, leads y
n 2 n 2 unloaded,n n 0 Passive x
to x?i? = v? as required, and, thus, to zero voltage x reflectors TX
n n − unloaded,n y
vn =0 at that port in the loaded case, when applying (4). (a) (b) TX
x
(c)
E. Optimization of the Load Resistance
Fig. 2. The three principles of WPT performance enhancement under con-
The presented optimization method provides optimal cur- sideration:(a)Shieldingusingpassiveelementsasreflectors,(b)guidingthe
magneticnear-fieldwithpassiverelays[11],and(c)mostgeneralenhancement
rents and reactive loading components, for a particular load
bymetasurface(or-material),e.g.,asproposedin[16],[17].
resistance R . This is a very practical case; as the load
L
resistance is usually not free to be chosen. However, in some Note: The aim is not to thoroughly investigate each en-
cases,particularlywheninvestigatingthemaximumachievable hancement technique, but to show the versatility of the afore-
performance of the WPT in general, also the optimal load mentioned optimization technique via these three relevant
resistance R is of importance. numerical examples.
L
To this end, the optimal load resistance can be obtained in
an outer optimization loop, as follows: A. Enhancement via Shielding by Passive Loops
R? =argminPrelax(R ) (22) One of the most straightforward ideas for WPT perfor-
L l,min L
RL mance enhancement is to try to confine the magnetic fields
Numerical experiments confirm the expectation, based on to the regions between the transmitter(s) and the receiver.
practical experience with dissipation of physical multiport This is often accomplished using a ferrite shield backing the
systems in general, that this remaining outer optimization of transmitter(s) and receiver(s) in directions away from each
finding the optimal load resistance is always convex. other [8]. However, due to the lossy nature of ferrites, their
Note that the optimal performance is generally not very application is usually restricted to the very low frequency
sensitive to the load resistance [27]. Thus, for practical con- regions. For mobile applications also their bulkiness, weight
siderations,theclosed-formR? foroptimalfullyactiveMISO and cost (or availability) can be an issue. Recently, using
L
IEEETRANSACTIONSONANTENNASANDPROPAGATION 6
resonant structures instead has been proposed, to reduce field PTE from about 13% to about 21% almost 26%, respectively.
leakage in unwanted directions [6]. Thus, adding passive loops—even at a greater distance to
The proposed test setups as shown in Fig. 3 are adopted the receivers—can lead to substantial enhancement of the
fromactiveMISOWPTsysteminvestigations[4]:oneandtwo performance. However, when comparing the currents and
additionaltransmitterloops,respectively,areplacedbehindthe power contributions of each of the loops, as listed in Tab. I,
first transmitter (thus at greater distance to the receiver), at a it is observed that the optimal operating conditions for these
constantseparationof∆z =λ/1000,whereλisthefreespace additional loops lead to currents comparable to the MISO
wavelength of the operating frequency f =13.56MHz. cases. Thus, when operated optimally, the additional loops
do not act as reflectors or shielding. Instead, they seem to
z z z
be used as additional transmitters, excited passively from the
r=λ/200 remaining active transmitter, but otherwise very similar to the
0
2
/ active MISO case.
λ θ θ θ
= Lastly, as Fig. 4(c) confirms, apart from areas of general
d y y y numericaldifficultiesintheanglesaround 60◦,thetightness
±
errorsareintherangeof10 14 to10 12.Thus,therelaxation
x x x − −
istight,meaning(14)isequivalenttotheoriginal(nonconvex)
(a) (b) (c)
program (8) and the one true global optimum has been found.
Fig.3. Thestandard(reference)SISOWPTsystem(a)andtwocaseswith
appended passive elements: (b) the SISO-P1 with one passive and (c) the
SISO-P2withtwopassiveelements,separatedat∆z =λ/1000.Operating
frequencyf =13.56MHz,wirethicknesst=5mm. TABLEI
CURRENTSANDPOWERCONTRIBUTIONSOFTHEACTIVEMISO-2/-3
SETUPSCOMPAREDTOTHEPASSIVELYENHANCEDSISO-P1/-P2CASES.
It has been demonstrated for MISO WPT systems that
such constellations can outperform the standard SISO setup MISO-2(twoactiveTX) SISO-P1(onepassiveelem.)
(a), when all transmitters are active and excited optimally. n in (inA) Pn (inW) in (inA) Pn (inW)
However, in this case, the additional loops are used as pas- 1 22.6∠91.3◦ 4.60 22.6∠91.4◦ 12.7
sive elements only, meaning they do not contain any active 2 22.2∠91.3◦ 8.12 22.2∠91.3◦ 0
excitation, but are excited via passive coupling only. The Total (η=7.86101%) 12.7 (η=7.86100%) 12.7
passiveloopsareshortedby(lumped)reactiveelementswhose
(a)
reactances are then optimized using the proposed method,
along with the receiver reactance, to obtain the optimally
MISO-3(threeactiveTX) SISO-P2(twopassiveelem.)
enhanced power transfer from the transmitter (black) to the n in (inA) Pn (inW) in (inA) Pn (inW)
receiver (blue). As mentioned, the load resistance RL in the 1 15.4∠91.4◦ 1.07 15.4∠91.5◦ 9.28
receiver could also be optimized, but since the impact on the 2 15.1∠91.4◦ 3.03 15.1∠91.4◦ 0
final performance is usually negligible, this is omitted, here. 3 14.8∠91.4◦ 5.18 14.8∠91.4◦ 0
The SISO WPT setups with one and two passive elements are Total (η=10.77904%) 9.28 (η=10.77903%) 9.28
called SISO-P1 and SISO-P2, respectively. (b)
Fig. 4 compares the resulting maximum PTEs of the ac-
tive MISO solutions (with multiple active transmitters) and
B. Enhancement via Passive Relays
the passive configurations in all three cases, along with the
tightness errors. Interestingly, the maximum achievable PTEs Another intuitive method to enhance WPT over larger
are very close (a); even relative differences (b) are very distances is placing a passive element in-between the trans-
small and only noticeable at angles where power transfer mitter and the receiver, so that it may act as a relay, over
is difficult in general. As can be seen, similar to the active which the power transfer takes place [10]. With multiple
MISO cases, adding one or two passive loops increases the such relays entire relay-line-type guiding structures can be
SISO MISO-2vs.
3=0 3=0 3=0 SISO-P1
MISO-2 SISO-P1
30 !30 30 !30 30 !30SISO-P2
MISO-3 MISO-3vs.
SISO-P1 SISO-P2
60 SISO!-P602 60 !60 60 !60
90 !90 90 !90 90 !90
2=0 10 20 30% "2/2 =!0.2 !0.1 0 log0=!16!14!12!10 !8 !6
(a) (b) (c)
Fig. 4. Performance comparisons (a) of the full (active) MISO (lines) and the SISO cases with passive elements (x markers): MISO-2 vs. SISO-P1 (blue)
andMISO-3vs.SISO-P2(red).Thedifference(b)isoftenalmostunnoticeable.Thetightnesserrors(cid:15)in(c)confirmthattherelaxationsinforbothsetups
(blueSISO-P1,redSISO-P2)areequivalenttotheoriginalproblemsandthattheglobaloptimahavebeenfound.
IEEETRANSACTIONSONANTENNASANDPROPAGATION 7
constructed [11], [12] and could potentially even facilitate number of relays leads to continued performance enhance-
power distribution over multiple paths and to multiple loads ment, but the increase in PTE slows down. Nevertheless, four
[13]–[15]. In addition to extending the high-efficiency range relays(f)leadtoalmost90%PTE,tenrelaysareabletoexceed
of the power transfer, relays can also help mitigate issues due 95%.Asthefieldplotsreveal,withaddingrelays,themagnetic
to misalignment between the transmitter and the receiver. field is more and more confined to the inside of this guiding
Bothpossibilitiesareinvestigatedinthefollowingexample, structure, forming a bent beam from the transmitter to the
starting off from the SISO setup as shown in Fig. 5(a): The receiver and reducing the overall outside fields considerably,
transmitterandreceiver(radiusr =r =λ/200)areplaced while the transferred power remains constant.
tx rx
at either ends of a quadrant of a circle of radius r = λ/20, The graph in Fig. 6(a) shows the optimal performances vs.
at the operating frequency f =13.56MHz. Even though they number of relays for different radii of the circular paths. The
are placed at an electrically small distance (d λ/14), the previously mentioned rapid improvement of the performance
≈
maximum achievable PTE of this system is only about 13% whenusingonlyasmallnumberofrelaysisonlyobservedfor
(b)—due to their severe misalignment. small radii; at greater distances a considerably larger number
Aspassiverelayelementsareinsertedintothecircularpath of relays is needed to achieve substantial performance en-
as shown in Fig. 5(c)-(h), the situation changes considerably; hancement.However,eventhere,eventuallythePTEincreases
higher optimum load resistances result and the overall perfor- substantially and seems to asymptotically approach a value
mance of the WPT system is enhanced considerably. As can close to 100%, only restrained by conduction loss of the
be seen, a single relay (c), leads to the largest performance relays. As expected, unloaded (open-connected) relays remain
enhancementstep:thePTEincreasesfrom13.2%toover60%. invisible (solid black line), whereas shorted relays would lead
Interestingly,thisisonlyabout9%underwhatcouldhavebeen to decreased PTE (dotted black line).
achieved when using the passive relay as active transmitter. The graphs in Fig. 6(b) again prove that the relaxations
Thus,thepassiverelaysetupisperformingconsiderablybetter were tight in all cases (with tightness errors in the range of
than a series of two SISO WPT systems of half the distance: 10−14 to 10−10) and that, therefore, the true global optima
(69.5%)2 47.9% < 60.2%. Moreover, using the relay, the were attained.
≈
optimumloadresistancemorethanquadrupled.Increasingthe Whereas the optimization of all reactive loading elements
(a) z (b) z (c) z (d) z
|B|dB
RX RX RX RX
passive
relays
0
2
/
λ
= y y y y
r
x x x x
TX TX TX TX
Setup SISO SISO-P1 SISO-P2
η=13.3% η=60.3% η=78.2%
R? =0.134Ω R? =0.79Ω R? =1.95Ω
L L L
(e) z (f) z (g) z (h) z
RX RX RX RX
y y y y
x x x x
TX TX TX TX
SISO-P3 SISO-P4 SISO-P7 SISO-P10
η=85.5% η=89.1% η=93.4% η=95.2%
R? =3.78Ω R? =5.97Ω R? =13.3Ω R? =20.4Ω
L L L L
Fig. 5. Comparison of some relay-enhanced WPT configurations, guiding power transfer along an arc (of radius r = λ/20). Adding relays enhances the
WPTperformance(max.achievablePTE)notonlybyconstrainingthefieldsalongthearc,butalsobymitigatingtheissueofmisalignment.Theperformance
increase of adding a single relay is the largest; inserting more relays improves the overall performance further and also continues to lead to larger optimal
loadresistances.
IEEETRANSACTIONSONANTENNASANDPROPAGATION 8
100 100
%
80
80 n
i
% R)E ηL 60 RL=RL?
ηPTE ηTEin 4600 r=λ/20 η(PTEPT 2400 rrr===λλλ///21500 RL=RLa
P r=λ/10 0
r=λ/5 10-1 100 101
20 relaysopen
relaysshorted DeDveiavtiaiotinonR RLL//RRLaLa
0 (a)
0 2 4 6 8 10 12 14 16 18 20
Number of Relays
NumberofRelays 102
(a) ΩR (in )RLL OAnpatilmytuicmRRLaL?
ε(cid:15)Norm. tightness error TightnessError1111100000-----1111864200 2 4 6 r=8 λ/2100 1r2=λ1/410 16r=18λ/5 20 *Optimal R and Analytic LoadResistanceL 111000-011 rrr===λλλ///21500
0 2 4 6 8 10 12 14 16 18 20
Number of Relays
NumberofRelays NuNmumbbeerr ooff RReelalaysys
(b) (b)
Fig. 6. Comparison of the performance enhancement due to optimal inter-
mediate passive elements (“relays”) for three distances (solid lines) and the %% 0.8
maximumPTEsiftheclosestrelaywouldbeusedasactivetransmitter,serving nn r=λ/20
awsitahninucprpeearsibnogunnudm(dbaesrhoefd)r.elTahyiss(ba)o.uTnhdecraenlabxeataiolmnsosatreartbigithratriinlyalalpcparsoeasch(be)d. ∆ηmentivement i∆η 00..46 rr==λλ//150
ovempro 0.2
(fionr athcehsieevicnagseospctiampaalciptievref)oromfanthcee,piatsissivneotresleanyssitiivsectroititchael ImprPTE I 00 2 4 6 8 10 12 14 16 18 20
receiver load resistance. This is addressed in Tab. II and NuNmumbbeerr ooff RReelalaysys
7, where the PTEs and loading capacitances are compared (c)
for cases with the optimum load resistance R? and with
L Fig. 7. Assessment of the importance of optimizing the receiver resistance
estimatesobtainedfromtheclosed-formexpressionsforactive RLforachievingoptimalperformance:ThemaximumPTEisawell-behaved
MISO WPT systems Ra. It can be seen that the difference concavefunctionofRL(a),well-suitedforoptimizationfromtheclosed-form
L estimateRa.However,eventhoughthedifferenceinloadresistancecanbe
in performance is small, even though the differences in the L
significant(b),thedifferenceinperformanceisusuallynegligible(c).
used load resistances can be rather large. The examples in
Fig. 7(a) underline the fact that the optimum PTE is rather
robust with respect to the actual R around the optimum in an outer loop around the presented semidefinite relaxation
L
valueR? andthatthePTEisawell-behavedconcavefunction with Ra as starting point. While the difference between the
L L
of the load resistance R , enabling numerical optimization optimum load R? and the estimate Ra grows larger with
L L L
TABLEII
COMPARISONOFPERFORMANCESANDREACTIVELOADSFORSISOSETUPSWITHUPTOFIVEPASSIVERELAYSWHENUSINGTHEOPTIMUMRECEIVER
RESISTANCEVS.USINGESTIMATESOBTAINEDFROMTHEACTIVEMISOCLOSED-FORMEXPRESSIONS.
Setup SISO SISO-P1 SISO-P2 SISO-P3 SISO-P4 SISO-P5
RL= RL? RLa RL? RLa RL? RLa RL? RLa RL? RLa RL? RLa
ηmax (%) 13.3 13.3 60.3 59.5 78.2 78.0 85.6 85.3 89.1 89.0 91.1 91.1
RL (Ω) 0.134 0.134 0.79 0.57 1.95 1.56 3.78 2.97 5.97 4.66 8.35 6.49
C? (pF) 104.43 104.43 104.38 104.32 104.35 104.29 104.20 104.02 103.98 103.69 103.68 103.27
r
C? (pF) 104.29 104.22 103.35 103.55 102.53 102.51 101.11 101.10 99.40 99.40
p
103.35 103.20 100.80 101.28 98.79 98.82 96.16 96.15
102.53 102.11 98.79 99.48 96.26 96.38
101.11 100.31 96.16 97.01
99.40 98.14
IEEETRANSACTIONSONANTENNASANDPROPAGATION 9
an increasing number of relays, as shown in Fig. 7(b), the The graphs in Fig. 9(a) present the fundamental results.
resulting difference in PTE is decreasing, plotted in Fig. 7(c). As reference and lower bound, the black dotted line and the
The resulting capacitances are in a practically realistic region markersdenotethecaseswithoutthemetasurface(allelements
of around a hundred picofarads, with the relay capacitances open-connected) or when all metasurface loops are shorted;
depending more heavily on the number of relays than the since the loops are electrically very small, the performances
receiver reactance. Furthermore, in the case with optimum of both of these cases are very similar. The most important
receiverresistanceR?,theresultingcapacitancevaluesappear result is the solid red line, giving the maximum achievable
L
to always be symmetric along the guiding path. performance with an optimum passive metasurface present.
Forcomparison,thedashedredlineplotsthePTEofanactive
C. Enhancement via General Metasurfaces and -Materials metasurface, where each loop of the metasurface is connected
Finally, the most general form of enhancement of wireless to a generator, serving as an upper bound on the achievable
power transfer is considered: Enhancement by metamaterials performance.
and metasurfaces. Such enhancements have been considered
bymanyresearchgroups,seeforexample[16]–[18].Forsim- 100
OptimumpassiveMS max
plicity and brevity, only an example with a metasurface (MS)
OptimumactiveMS open
willbeconsidered;thegeneralizationto(bulk-)metamaterials 80 MSelementsopen(invisshibolert)
avg
is straightforward. Instead of using any direct methods (e.g. MSelementsshorted
Huygens surface approaches using the equivalence principle) in% 60 MMSSuunniiffoorrmmllyyllooaaddeedd((34pw..91ao80sr98ssnntivFFe))
to determine the surface impedance and obtain the required η
metasurface elements from that, the approach here is to TE 40
P
optimize the loading of all of the elements of a particular
metasurface with the semidefinite-relaxation based method 20
presented herein.
Fig. 8 shows the setup of the metasurface-enhanced WPT 0
0.02 0.04 0.06 0.08 0.1
example under consideration: The transmitter and receiver
Distance d/λ
loops (of diameters D = D = λ/100) are separated by
tx rx
distance d. In the middle, a 15 15-element planar loop array (a)
×
(loop diameters λ/500, arranged periodically at a uniform
(cid:15)
pitch of λ/200 in the x- and y-directions) is inserted, which or 10-11
is referred to as metasurface. Each of the loops is loaded Err
by a reactance (typically a capacitor Cn,m), which is to be ss 10-13
e
optimized for maximum power transfer efficiency together n
ht
with the receiver reactance x . The closed-form expression g 10-15
r Ti 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
(25) is used to obtain the receiver resistance R .
L Distance d/λ
(b)
z
00 λ/500 xrRL RX FOipgt.im9.umCoamctipvaeriasonndpoafsPsiTvEesPoTfEtshe(dmasehteadsuarnfadces-oelnidharnecde,dreWspPecTtivsyelsyte),mo:p(ean)
/2 and shorted metasurface loops (black), and two cases of uniform loading: a
λ
“good average” (blue) vs. the worst-case (green). The tightness error of the
d/2
optimumpassivemetasurface(b)provestightnessoftherelaxation.
d/2
As canbe seen, the enhancementdue to thepassive surface
15M×et1a5s-uErlfeamceent TX y is significant: For example at distance d=0.05λ, the passive
surface leads to an additional 20% in PTE over the reference
x
case without a metasurface. On the other hand, at the same
distance,theperformancewiththepassivemetasurfaceisonly
Fig.8. SetupoftheWPTsystemenhancedbyapassive15 15-elementloop-
× about 10% behind that with an active metasurface.
basedmetasurface(MS).Dimensions:transmitterandreceiverloopdiameters
Dtx = Drx = λ/100 ( 44cm), wire thicknesses λ/2000 ( 2.2cm) The blue and green solid curves in Fig. 9(a) address the
for transmitter and receive≈r and λ/104 ( 4.4mm) for metasurf≈ace loops, importance of adequate loading of the metasurface elements.
≈
respectively. Distance d = 0.01...0.1λ ( 0.44...4.4m) at the operating
≈ In each of these two cases, all loops of the metasurface are
frequencyoff =6.78MHz.
loadeduniformlybythesamecapacitance:about3.989nF(the
This case can be optimized in straightforward manner with optimal center capacitance at d = 0.06λ) in the blue case
the presented method. The 225 elements of the 15 15 loop and 4.108nF in the green case. The blue curve represents a
×
surface in this case lead to a problem size of over 200000 case of good overall performance, which after some distance
unknowns,sincethequadraticcurrentmatrixChas(2 (225+ (about 0.04λ) is largely comparable to the optimum (solid
×
2) 1)2 entries. However, the problem is still solvable within red). The green curve represents a worst-case scenario: Using
−
a moderate computation time of a few minutes (on an Intel i3 anonlyslightlydifferentcapacitancevalue(differingbyabout
machine running CVX/SDPT3 in Matlab). 3%),degradesthemaximumoverallperformancefarbelowthe
IEEETRANSACTIONSONANTENNASANDPROPAGATION 10
casewithoutametasurface(thereceiverreactanceisoptimized receiver, the larger the variance between different optimum
to still maximize the performance using the suboptimally capacitancevalues:theradialwave-likechangeincapacitance
loaded metasurface). Hence, the loading of the metasurface from the center of the metasurface to the outside at 0.01λ
is absolutely crucial for optimal performance enhancement. changes into a convex profile at 0.04λ and eventually into a
concave profile at 0.08λ and further away. Edge effects due
z z to the finite dimensions of the metasurface are responsible for
B
| |dB theoptimalvaluesneverbeingthesamethroughoutthewhole
area.
RX RX
V. SUMMARY&CONCLUSION
The performance of WPT systems is limited by the size of
the whole system. The requirement of high power transfer
efficiency (PTE) generally puts a hard constraint on the
distance between transmitter(s) and receiver(s); usually this
distance can only be a small fraction of the wavelength at the
y y
operating frequency and is comparable to the transmitter loop
x x size.
TX TX
Manyenhancementshavebeenproposedtoimprovetheper-
formanceoftheWPTsystem.Oneofthemoststraightforward
is to place passive elements in the proximity of the trans-
(a) (b)
mitter and/or receiver. Rigorously optimizing such systems
Fig. 10. The magnetic fields of the standard SISO system (a) and the is not a trivial task, since the required power constraints are
SISO-P225 system (b), enhanced by an optimally loaded 15 15-element
× nonconvex, thereby rendering the entire optimization problem
passive metasurface. The transmitter and receivers are located at a distance
of d = 0.08λ, while the metasurface is placed in the middle, at d/2. The unsolvable in general, at least for a large number of passive
maximumachievablePTEsareabout1.2%forthereference(a)and8.6%for elements.
themetasurface-enhancedcase(b).
A fully equivalent work-around via tight semidefinite re-
ThegraphinFig.9(b)confirmsthattherelaxationtoobtain laxation has been presented, which solves this optimization
the maximum PTE with the passive metasurface (red solid) problem efficiently and reliably for an arbitrary number of
was tight in all cases under consideration, as the tightness passive elements. Starting from the impedance matrix of the
error is in the range of 10 14 to 10 12. whole WPT system and the load resistance of the receiver,
− −
Fig. 10 shows a comparison of the magnetic fields of the the maximum achievable PTE as well as the required optimal
standardSISOWPTsystemandwhenitisoptimallyenhanced currents and reactive loading elements are found. In an outer
by the 15 15-element metasurface. Since the receiver load optimization loop, the load resistance can also be optimized;
×
resistances differ very little and unit power is transferred in however it is generally observed that the optimum deviates
bothcases,thefieldsaroundthereceiverloopsareverysimilar. verylittlefromtheoneobtainedfromMISOoptimizationand
However, the fields around the transmitter are quite different, that the maximum PTE is not sensitive to small deviations. A
since in the enhanced case the efficiency was improved from simpletestofthesolutionsconfirmstightnessoftherelaxation;
just above 1% to about 8.6%, requiring much lower input i.e.thatthesolvedproblemremainedequivalenttotheoriginal
power and reducing the field amplitudes accordingly. one and that the true global optimum was found.
Lastly, in Fig. 11, the optimum load profiles of the meta- With this powerful and versatile optimization method at
surface elements are compared for three distances d. As can hand, three examples of passively enhanced WPT systems
be seen, the closer the metasurface is to the transmitter and have been investigated. It has been shown that optimally
dd d= = =0 0 .00.0.101λ1λλ dd d= = =0 0 .00.0.404λ4λλ dd d= = =0 0 .00.0.808λ8λλ
Cn?,m (nF) d = C0.0n?1,λm (nF) d = 0.04λCn?,m (nF) d = 0.08λ ∆Cn,m
0.1
555 555 555
44.45.5.5 44.45.5.5 44.45.5.5
5 5 5 0.05
444 444 444
4.5 4.5 4.5
33.35.5.5 4 33.35.5.5 4 33.35.5.5 4
0
333 3.5 333 3.5 333 3.5
22.25.5.5 3 666 22.25.5.5 3 666 22.25.5.5 3 666
444 444 444 -0.05
222 2.5 002022 222 6 2.5 002022 222 6 2.5 002022 6
666 444 22n2x000-2--22-4--44-6--6266 -46---646---424-2-22n0y -2 -4 -6666 4-464-4-220222n4x000-2--22-4-2-446-6--664 2-6---646---4024--22-2ny-4 -6 -666-64-2044244222n0x00-22--226 -4-4-442-6--606-2--46---64-6-6--424--22 -n6-4-y2024 -0.1
(a) (b) (c)
Fig.11. Optimalcapacitanceprofilesforthe15 15-elementmetasurfaceatdistances(a)d=0.01λ=Drx,(b)d=0.04λ=4Drx,(c)d=0.08λ=8Drx.
×