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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 9, SEPTEMBER 2014 4623

Convex Optimization of Wireless Power TransferSystems With Multiple Transmitters

Hans-Dieter Lang, Student Member, IEEE, Alon Ludwig, Member, IEEE, and Costas D. Sarris, Senior Member, IEEE

Abstract—Wireless power transfer systems with multiple trans-mitters promise advantages of higher transfer efficiencies andfocusing effects over single-transmitter systems. From the stan-dard formulation, straightforward maximization of the powertransfer efficiency is not trivial. By reformulating the problem,a convex optimization problem emerges, which can be solvedefficiently. Further, using Lagrangian duality theory, analyticalresults are found for the achievable maximum power transferefficiency and all parameters involved. With these closed-formresults, planar and coaxial wireless power transfer setups areinvestigated.

Index Terms—Convex optimization, maximum transfer effi-ciency, multiple transmitters, wireless power transfer.

I. INTRODUCTION

W IRELESS POWER TRANSFER (WPT) continues togain much attention ever since Tesla’s ideas [1] were

rediscovered, realized, and extended a few years ago [2]–[4].Besides many applications, as well as the fundamentals, in par-ticular the limits of the transfer efficiency and enhancementsthereof have been studied [5]–[7].More recently, systems with multiple transmitters were in-

vestigated, both theoretically and experimentally [8]–[10]. Mul-tiple transmitters provide more degrees of freedom of the pri-mary field or current distribution and, therefore, promise thepossibility of enhanced transfer efficiency as compared withsingle-transmitter systems. In this paper, systems with multipletransmitters and a single receiver are investigated. In this con-text, they will be calledMISO (for multiple-input single-output)systems, similar as, for example, in communication systems. Incontrast, systems with only one active transmitter and a singlereceiver are called SISO (single-input, single-output) WPT sys-tems.Optimization of WPT systems with multiple transmitters

(MISO WPT systems), particularly the required excitations(voltages) and the capacitive loading to maximize the achiev-able power transfer efficiency, is not trivial. Numerous testshave shown that even more elaborate and generally powerfulglobal optimization algorithms, such as pattern search andgenetic algorithms as well as particle swarm optimizers, have

Manuscript received January 03, 2014; revised April 29, 2014; accepted May31, 2014. Date of publication June 12, 2014; date of current version September01, 2014.The authors are with Electromagnetics Group of the Department of Elec-

trical and Computer Engineering, University of Toronto, ONM5S 3G4, Canada(e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2014.2330584

problems finding the best parameters—or they lead to infeasiblecomputational cost and long optimization periods. In addition,using these methods, physical insight into the problem and itscharacteristics is not necessarily available in a straightforwardmanner but usually can only be obtained by backtracking theend results.In this paper, a physically intuitive design procedure using

convex optimization is presented. Besides being able to obtainresults such as the maximum power transfer efficiency and therequired excitations and capacitive loading efficiently, each stepof the formulation is physically meaningful. Furthermore, usingLagrangian duality theory, analytical solutions for all parame-ters involved are found. Applied to theWPT systemswith one ormore transmitters, various interesting conclusions can be drawn,for example in relation to subwavelength focusing [11]–[13].It is well understood that impedance matching of the receiver

load to the total WPT system as source is important to achievemaximum transfer efficiency [14]–[16]. However, althoughit might be feasible to tune capacitors during operation (e.g.,using varactors), adjustment of the load resistance via (tunable)matching network is difficult and results in additional loss.Therefore, the general question to be answered within thispaper is, “What is the maximum achievable power transferefficiency for a particular WPT geometry/setup to a loadand how can it be achieved?” In addition, by analyzing theclosed-form expression of the maximum transfer efficiencywith respect to , the optimum as well as the maximumachievable power transfer efficiency are found.The outline of this paper is as follows. First, in Section II,

the problem setup is introduced and the general mathematicalrelations are given. Second, in Section III, the convex optimiza-tion procedure is derived. In Section IV, using Lagrangian du-ality theory, analytical solutions to the optimization problemsare formulated for all parameters involved. Section V discussesand compares some basic results and performances of MISOand SISO WPT systems. Finally, in Section VI, a summary andconclusions are given.

II. PROBLEM STATEMENTA remark on the notation: Italicized letters represent vari-

ables, bold small letters refer to vectors, bold capital letters arematrices (and similarly for functions resulting in vectors andmatrices); stands for the transpose of the vector , while

stands for its Hermitian (conjugate transpose). Real partsof complex numbers are denoted by , imaginary parts by

, and the conjugate complex is denoted by , not to beconfused with the optimum solution to a particular opti-mization problem.

0018-926X © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

4624 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 9, SEPTEMBER 2014

Fig. 1. Illustration of the problem setup: multiple transmitters (here three loopsto ), at the positions and

one receiver at . Eachloop is loaded by a capacitor ; the receiver end also contains the resistiveload .

A. System Geometry and Electrical Setup

The WPT systems considered here consist of trans-mitter loops, which are located at positions ,for (where stands for the total number oftransmitters, the total number of loops is ) andone receiver loop, located at , as shown in Fig. 1.Each loop is loaded with a capacitor (corresponding to

the respective capacitive reactance of interest at the operatingfrequency). In addition, the receiver loop is loaded with the re-sistive load . Each transmitter loop is excited by a voltagesource .Note that, although here only cases with aligned (parallel)

loops are considered, this is not a requirement. In fact, thederivation is general and can be used for impedance matrices ofarbitrary power transfer systems with one or more transmittersand a single receiver.The idea is to assess the maximum achievable power transfer

efficiency of a givenWPT system in a given direction. Thus, thereceiver loop is positioned at various points in space, scanningover a region around the transmitter loops. At each position,the capacitors and the excitations are to be optimized so as tomaximize the total power transfer efficiency .

B. Initial Problem Formulation

Using a voltage vector and a currentvector (all ), where the last( th) entries correspond to the receiver loop, the full systemcan be formulated as

(1)

where , with and , isthe unloaded impedance matrix of the WPT setup and

is the fully loaded impedancematrix, to be optimized at a later point. The capacitive loading

comes from , a diagonal matrix containing the(real) reactances of the capacitors

(2)

where all and are real variables, and is the angularfrequency of operation. Note that the last reactance value is de-noted by instead of to simplify the notation throughout thederivation later on. Finally, the load matrix

.... . .

...... (3)

is zero everywhere, but at the very last element, correspondingto the receiver; denotes the (real) load resistance at thereceiver end.In more detail, the total system is

...

......

. . ....

...(4)

where denotes the elements from the unloaded impedancematrix . Note that , since the load is part of theloaded impedance matrix .The goal is to find the optimal capacitive loading and cur-

rents so as to maximize the total transfer efficiency, given by

(5)

where

(6)

and

(7)

under the constraint that the actually transferred power isnonzero, or more precisely larger than some threshold value,

.

LANG et al.: CONVEX OPTIMIZATION OF WIRELESS POWER TRANSFER SYSTEMS WITH MULTIPLE TRANSMITTERS 4625

Fig. 2. Typical result of optimizing the power transfer efficiency directly fromits standard formulation (real and imaginary parts of voltages, capacitor values)using genetic algorithm (population size: 50, maximum number of generations:1000). The resulting efficiency pattern is spiky and asymmetric (even thoughthe geometry is symmetric).

Fig. 3. Illustrations of the planar MISO WPT setups: multiple transmitters,with the position and a receiver at :(a) SISO (reference); (b) MISO-21; and (c) MISO-31.

Various tests have confirmed that it is difficult to optimizefrom (1), (6), and (7) directly, for example using global op-

timization methods such as genetic and pattern search algo-rithms as well as particle swarm optimizers. Often, the resultsvary greatly between receiver positions arbitrarily close to eachother (e.g., between points at distances much smaller than aquarter-wavelength), or they are unsymmetrical, even thoughthe problem geometry is symmetric, as shown in Fig. 2 (seesetups in Fig. 3). These numerical artifacts originate from both

being a nonanalytic operation and the nonconvex (as wellas nonlinear and highly resonant) form of (5) using (1), (6), and(7) in general.

III. FORMULATION OF A CONVEX OPTIMIZATION PROBLEMFOR MAXIMIZING MISO WPT EFFICIENCY

Before the convex optimization problem can be derived, thevoltages and currents are put into a vector form of separated realand imaginary parts to obtain a fully real-valued formulation.The total input power over all transmitters is calculated by

(8)

where and are real vectorscontaining the separated real and imaginary parts

of the currents and voltages, respectively.

In similar fashion, for the output power, it is obtained that

(9)

where

(10)

Using these expressions, the efficiency can be given in stan-dard form, involving only real-valued unknowns

(11)

where the (separated) currents and voltages are related by

(12)

similar to . Note that at this point only the capacitorreactances on the diagonal of and the voltages are un-knowns; the currents follow from (12).

A. Convex Optimization Formulation

In the following, the problem is reformulated using the cur-rents instead of the voltages . Combined with the assump-tion of unit received power, a convex optimization problem isobtained.1) Implicit Voltages: The voltages in the denominator of

(11) are replaced by their relation to the currents (12). From theloaded impedance matrix , it follows for (6) and (8),since , that

(13)

where is used to denote the matrix of the real parts of(i.e., whereas , so relates simi-

larly to ).The power transfer efficiency is then given by

(14)

which only contains as unknowns, since the capacitorvalues are not part of . However, as will be shown later,one of them is part of the equality constraints of the optimiza-tion.It should be noted that this form of is not generally concave

(or not convex), and it does not account for the zero voltageat the receiver, , yet.2) Convex Reformulation of the Objective Function: By set-

ting the received power to a constant ,


meaning that 1 W power is received at the receiver load, the ob-jective function for minimization according to

(15)

(with , since ) becomes

(16)

and is convex in , provided that (positive semi-def-inite), which is always true (apart from some numerical issues,addressed in the Appendix A) for impedance matrices of pas-sive networks [22]. In fact, since WPT systems are always lossy(there is at least some small loss due to radiation), the real part ofthe impedance matrices has to be positive definite,

, which also leads to . Thus, this objectivefunction is well suited for convex optimization (minimization).3) Constraints: As mentioned, the constraints have to ac-

count for the voltage at the receiver port being zero, since theload is part of the impedance matrix, as shown in (4). Thus,

(17)

with being the th (last) row of the loaded impedancematrix , corresponding to the receiver voltage . Sepa-rating real from imaginary parts, this translates to

(18a)(18b)

In addition, one is free to chose the real and imaginary parts ofthe receiver current , as long as the receivedpower remains . Thenatural choice is (zero phase at the receiver), and thus

(19a)

(19b)

which will have to be accounted for in the constraints during theoptimization process.These four constraints, (18a) and (18b) as well as (19a) and

(19b), are then cast into the matrix form

(20)

which is well known to be affine in [17], whereand . Note that all constraints for this problem are ofthe equality type; there are no inequality constraints.

In more detail, the first two constraints in (20) are

(21a)(21b)

where and . Thesecond pair of constraints are

(22a)(22b)

where and are row vectors selecting the correspondingth and th (last) entry, respectively, from the (real) current

vector .4) Partially Convex Optimization Formulation: Using the

objective function, (16), and the constraints, (20), a quadraticprogramming (QP) problem [17] is obtained:

(23)

Even though the reactance values are notpart of the convex optimization problem, the receiver reactance(the last entry in ) cannot be optimized this way, since it is

part of the equality constraints . Thus, the resultis the maximum transfer efficiency given the receiver reactance

.The problem of finding the maximum achievable transfer ef-

ficiency can be solved by choosing some receiver reactance ,optimizing by finding the best currents and then going backand choosing a better , resulting in an outer optimization loopfor :

(24)

5) Fully Convex Optimization Formulation: Using theknown receiver current components , givenfrom (19a) and (19b), explicitly (thereby reducing by twoentries, leading to the transmitter current vector ), the con-straints become

...

...

(25)

where as always , for MISOWPT systems. The re-ceiver reactance is an unknown and is, therefore, added to theunknown vector, whereby becomes , the modified currents


vector (losing some of its physical meaning, since it containsnot just currents anymore), as follows:

...

...

(26)

From here on, hats mark all “modified” vectors and matrices(i.e., such with components/conditions of the reactance addedto the ones for the currents and not including the known re-ceiver currents) and and refer to the real and imaginaryparts, respectively, of the unloaded impedance matrix , cou-pling each of the transmitters to the receiver [the first ele-ments of the th (last) row of ].The matrix of the modified affine constraints is then of dimen-

sions , since it only has to account for the transmittervoltage to be zero. This way, the receiver reactance can be opti-mized together with the transmitter currents, simultaneously, ina single optimization. The problem is still of the quadratic pro-gramming type, but it has to be adjusted somewhat, since thereceiver power and mutual parts of the receiver and transmittersare missing in the quadratic term:

(27)

where the modified matrix of the quadratic term is

... (28)

with being the part of the resistance matrix corre-sponding to the transmitters (i.e., reduced by the last columnand row). Note that this part of the matrix is the same for bothand (since ).The linear term is (since all resistance matrices are sym-

metric)

(29)

and the power in the receiver loop is(both loss and power dissipated in the load).Thus, the final single quadratic programming problem to op-

timize both currents and the receiver reactance such as to max-imize the transfer efficiency (or rather minimize its inverse) is

(30)

where, again, the objective is convex in since andthe equality constraints are affine. Clearly, the last added scalar

, the power in the receiver loop and load, is not necessary foroptimizing the parameters but is added to get the optimal valueto be , directly.Note that is only positive semi-definite and not

positive definite anymore, since one of its eigenvaluesis zero at all time (due to the row and column full of zeros).Further, note that is obviously not square; thus, the systemof constraints cannot just be solved linearly for and be put intothe minimization to find the solution (trivial case).

B. Transmitter CapacitorsAs derived, the maximum transfer efficiency, (24), is inde-

pendent of the transmitter capacitors (or rather their reac-tances ), , since it depends on the currentsin the loops (and the receiver reactance ) directly. However,the transmitter capacitors are essential to ensure that these op-timal currents actually result from the voltages, by which thewhole system is excited. Thus, in this second step, the correcttransmitter capacitor values to accomplish that task have to befound.Recall that a MISO- setup (total number of loops

) can be described as

.... . .

......

... (31)

Separating the voltages into real and imaginary parts andputting it into vector form leads to

(32a)

(32b)

for (but not , since the receiver voltage isalready contained in the constraints of the convex optimizationprocess to get the efficiency), where and are the real andimaginary parts, respectively, of the th row of the unloadedimpedance matrix .There is no requirement as to how the voltage and capac-

itor values should be chosen (other than practical realizability).From common (SISO-type) WPT systems, it is known that boththe transmitter as well the receiver should be driven at resonanceto maximize both the efficiency as well as the actual powertransferred to the load [7]. Since both the input and output of thetransfer system are of the series resonant circuit type, they aretuned to series resonance (in the coupled case, not separately),thus to the point where the impedance has its minimum and theamplitude of the voltage to produce a certain current is minimal.


To obtain similar results and because it is well known thatunconstrained least-square problems have analytical solutions,the capacitive reactance values (for ) arechosen so as to minimize the voltages (i.e., thereby tuning thecoupled transmitters to resonance)

(33)

Finally, the capacitor values are found using .

C. Summary of the Convex Optimization Formulation

Two algorithms to optimize both the currents and receivercapacitance such as to maximize the transfer efficiency of WPTwith multiple transmitters have been presented: The first is in aphysically intuitive inner/outer optimization loop form (whereonly the inner loop is a standard convex optimization form), andthe second is a fully convex optimization formulation.The optimum transmitter capacitors are not essential for max-

imizing the achievable power transfer efficiency. However, theyare responsible to generate the optimal currents from given ex-citation voltages. Here, they are chosen by analytically mini-mizing the excitation amplitudes for the required currents using(32b) and (33). Minimizing the excitation amplitudes relates theMISO solution to the maximized output power condition (trans-mitter resonance) for SISO WPT systems.

IV. CLOSED-FORM RESULTS FOR THE MAXIMUM TRANSFEREFFICIENCY OF MISO WPT SYSTEMS

The previously discussed optimization problem can be effi-ciently solved using well-known quadratic programming algo-rithms, and the solutions are perfectly smooth (no optimizationissues as experienced with the global methods). However, asshown next, this optimization problem actually has analyticalsolutions.As will be derived step by step, for WPT systems with mul-

tiple transmitters, all parameters, i.e., all currents and capacitors,can be found analytically such as to maximize the transfer effi-ciency, which has a closed-form solution as well. Finally, usingthat formulation, the load resistance can be optimized as well soas to find the maximum achievable transfer efficiency of a givenWPT system directly from its (unloaded) impedance matrix.

A. Dual Formulation

The basic idea of Lagrangian duality is to account for the ob-jective and the constraints in one function by adding a weightedsum of the constraints to the objective function [17]. If certainconditions apply, then, by optimizing these weights (called dualvariables), the problem can be solved using this dual formula-tion, which can be advantageous in some cases.The Lagrangian function [17] follows straightforwardly

from the convex optimization formulation, (23), by adding aweighted sum of the constraints to the objective function

(34)

where the dual variables (also called Lagrangian multipliers) forequality constraints are . Note that these variablesare unconstrained, i.e., . Further, note that if thesevariables are maximized (go to infinity), the original problemresults in the following:

(35)

Thus, the original problem is the minimum of the upper boundof the Lagrangian function.The dual function [17] is defined as the lower bound of the

Lagrangian given by

(36)

which in this case can be found analytically, at the point wherethe derivative with respect to the modified current vector (gra-dient) vanishes:

(37a)

(37b)

However, , as given in (28), is obviously non-invertible.A closer look at this problem reveals

(38a)

(38b)

The last element/row is only zero if ; thus, thesecond dual variable has to be , for the Lagrangian tohave a minimum.A dual variable equal to zero means there is slack in that

constraint or, in other words, that constraint is “not essential” tosolve the whole problem. Here, the optimal has no influenceon the rest of the problem (since the gradient of the Lagrangiandoes not depend on it) and can be obtained at a later point, usingthat particular constraint and the optimized currents.The transmitter currents that minimize the Lagrangian are,

therefore,

(39)

and the dual function (36) reduces to

(40a)

(40b)

This dual function is concave in if

(41)


is concave in , meaning . Substituting thefirst constraint vector in and using from(28), it can be seen that

(42a)

(42b)

since , for real passive impedance matrices, because ofradiation loss. Thus, the dual function is indeed concave inand, therefore, has a single global maximum.The theorem of strong duality (SD) [17] states that, if the

original problem is differentiable and convex in (which is truein this case, as stated before), then the minimum of the upperbound (the original problem) and the maximum of the lowerbound (dual problem) are equivalent:

(43a)

which translates to

(43b)

Thus, solving (minimizing) the dual problem is equivalent tosolving (maximizing) the original problem, in this case.Note that maximizing (40b) is still a quadratic programming

problem, as it should be, since QPs are self-dual. However, un-like the original problem, on the right-hand side of (43b), thedual formulation (left-hand side) is unconstrained, since .

B. Analytical Solution of the Dual Problem

Quadratic programming (QP) problems are known to haveanalytic solutions in some cases, especially unconstrained. Inthis case, the dual formulation of the QP problem has an ana-lytical solution. At the maximum of , the gradient with re-spect to the dual variable , which here reduces to just the partialderivative with respect to , vanishes:

(44a)

(44b)

As previously stated, and thus all dual variables are de-fined and the dual problem is solved. Using these solutions, thequantities actually looked for, the optimal transmitter currents

and the receiver capacitance (via its reactance) as well asthe transfer efficiency are derived in the following.1) Transmitter Currents: Substituting the optimal dual vari-

able from (44b) as well as the (impedance) vectorsand back

into (39), the transmitter currents are obtained as follows:

(45)

where the coefficients are given by

(46a)(46b)

2) Receiver Reactance and Capacitor: The optimal receiverreactance is obtained from the (up to this point still unused)second constraint:

(47)

Rearranging and substituting the transmitter currents , (45),and the dual variable , (44b), in leads to

(48a)

Finally, by replacing the vectors by their respective definitionsand as

well as using the fact that , the optimal receiverreactance is given by

(48b)

and the capacitor value is obtained using .3) Optimized and Maximum Achievable Transfer Efficien-

cies: Starting from the dual function (40b) and substituting inthe transmitter currents (39), the transfer efficiency (to a specificload ) can be expressed by

(49)

It might seem like the efficiency could turn negative becauseof the negative sign in the first factor in the denominator. How-ever, it can be shown that , becauseit is a Schur complement of the (symmetric) resistance matrix

. As such, it has to be positive definite (or in this case,just positive, since it is a scalar in this case) together with thesubmatrix being positive definite, for the resistancematrix to be positive definite, . Both of these last condi-tions are a given for lossy passive networks. Thus, the transferefficiency can never become negative.It can easily be seen that the transfer efficiency given in

(49) is concave in . Its maximum is found at

(50)

and, thus, the optimal load resistance is given by

(51)

Note that, for the same reason as before (Schur complement),the optimal exists for all real and passiveWPT systems (andcannot turn complex). Further, in the SISO case with identical


Fig. 4. Planar MISO-21, MISO-31 versus SISO: maximum transfer efficiency for and vs. and optimum load resistance , forand lossless, , (a), (c) and lossy, , (b),(d) loops at a distance . (a) Transfer efficiency , thick, , thin, for ,

(b) Transfer efficiency , thick, , thin, for , (c) Optimum load resistance (in ), for , (d) Optimum load resistance (in ), for.

antennas , (51) and (49) lead to an iden-tical to the one given in [5].Using the optimal load (51) in (49), the maximum achievable

transfer efficiency of any MISO WPT system can be obtained.As expected (since the impedance matrix actually already con-tains all information about the WPT system), the maximum per-formance only depends on the systems (unloaded) impedancematrix.

V. RESULTS

A. Problem SetupsFor this paper, the unloaded impedance matrix was ob-

tained using the Double Multiradius Bridge-Current (DMBC)method, [18], [19], implementing a thin-wire, piecewise sinu-soidal Galerkin moment method in 64-bit precision. DMBC isknown to be powerful and accurate, [20], [21].For verification, the optimized values were included in loaded

models and simulated using DMBC again. The results werecompared and agreed very well. Differences are mainly due tothe fact that for DMBC the capacitances and excitation voltageshave to be handed over in a text file and, thus, the values are sub-ject to round-off errors.Depending on the method or software used to obtain the un-

loaded impedance matrix, and for some models, non-passiveimpedance matrices might be received. Using DMBC, this wasthe case for a few angles, when considering lossless or veryhighly conductive (but finite) loops (e.g., ) andparticularly for WPT systems with a higher number of ports,e.g., seven loops (i.e., MISO-61) or more. Since in these casesthe received impedance matrix does not represent the physics

of the model, passivity has been enforced as described in theAppendix A. The change in the norm of the impedance matrixalways remained negligible.

B. Planar Setups1) One-Dimensional Models: First, the transfer efficiency of

WPT systems with multiple (up to three) transmitters is inves-tigated. Fig. 3 shows the setups in question: the MISO-21 andMISO-31 as well as the SISO reference. The transfer efficiencypatterns for as well as the optimum load resistance

are shown in Fig. 4, for loops with infinite conductance ( ,left) and copper conductance ( S/m, right), alongwith the respective optimum patterns.The geometrical parameters involved are as follows:• separation between the transmitter loops ;• loop radii ;• wire thickness

where and 40 MHz (the results are scalable).As expected, the maximum achievable power transfer effi-

ciency is higher in the cases with lossless loops than in thosewith lossy loops. However, it is more than just an overalldrop; for cases with multiple loops, the overall form of pat-tern changes. As is well known, impedance matching, meaningusing the optimum load , has a strong impact on the achievedtransfer efficiency, particularly at broadside.Note that, for lossless loops, the MISO-21 setup can perform

somewhat worse than the SISO case at broadside, since it ismissing a center element, whereas the MISO-31 always has toperform equally well or better than the SISO reference.2) Two-Dimensional Models: Fig. 5 shows two-dimensional

setups with loops arranged the -plane (at a constant distance


Fig. 5. Illustrations of the larger MISO WPT setups; the satellite transmittersare arranged symmetrically in the -plane, at a constant distance ofaround the center transmitter position. (a) MISO-51 setup and (b) MISO-61 andMISO-71 (including the dashed center element) setups.

Fig. 6. Transfer efficiencies for MISO-51, -61 and -71 setups versus MISO-31,without and with conduction loss: while in the lossless cases there is substantialefficiency gain from using this many loops, in the lossy case all patterns collapseto the MISO-31 pattern. (a) , thick, , thin, for , and

, (b) , thick, , thin, for , and .

to the center element of ). Their transfer efficiency pat-terns for are shown in Fig. 6. The results seem physically rea-sonable, by comparison, since the resulting efficiency patternfor the MISO-71 setup is the sum (“overlay”) of the patternsof the MISO-61 (no center element) and MISO-51 (center ele-ment) setups.While the patterns in Fig. 6(a) reveal substantial transfer effi-

ciency enhancement using multiple loops even at a distance of, in the lossy case, see Fig. 6(b), all efficiency patterns

collapse more or less to the one of the MISO-31 setup, no matterif the optimum load is present or not.3) Comparison and Interpretation: Comparison of the loss-

less and lossy cases reveals that the overall working princi-ples are different, for the two cases. Without conduction lossin the loops, the only loss to minimize (to maximize transfer ef-ficiency) is radiation. Minimizing radiation is achieved by min-imizing far-field effects using (almost) canceling primary fields

due to currents of opposing direction and almost equal ampli-tudes, see Fig. 8(a). Since the amount of transferred power isconstant 1 W, and the currents almost cancel each others fields,the primary currents are very high. Additionally, these currentsand fields do not change substantially, for different receiver po-sitions, shown in Fig. 7.In cases with conduction loss, such high currents are not fea-

sible and the far-field cancellation cannot be achieved, at leastnot to the same degree, as shown in Fig. 8(b). The optimal cur-rents in the transmitter loops are very different for different re-ceiver positions—essentially all the energy is put into the ele-ment closest to the receiver, shown in Fig. 7(b). The primaryreason for higher transfer efficiency is closer proximity; thetransmitter closest to the receiver is favored, while the otherscarry much lower currents.This is the reason why there is no (substantial) performance

enhancement for lossy setups with more than three transmitters,as shown in Fig. 6(b): Since the distance to the closest trans-mitter remains the same (for receivers in the -plane) for allthese setups, so does the achieved transfer efficiency. On theother side, in the lossless cases, the far-field cancellation can beachieved more efficiently (particularly in some directions) witha larger number of transmitters, leading to an increased transferefficiency, as can be seen in Fig. 6(a).4) Verification: This is verified using the models shown in

Fig. 9, where one or two loops are positioned off-center so asto represent part of the (centered) MISO-31 setup. As can beseen from the transfer efficiency patterns in Fig. 10, in the lossycase (b) the final MISO-31 pattern can be obtained by “adding”the patterns from the offset cases, thus confirming that the setupwith multiple transmitters is essentially the same as a single loopat the position closest to the receiver.Note that this is not the case for lossless loops: Due to the fact

that the fields can be canceled more efficiently with more loops,the transfer efficiency pattern of the MISO constellation cannotbe obtained by overlaying the separated patterns.5) Realistic Capacitor Values: It is worth noticing that the

obtained optimal capacitor values are not unrealistic for a prac-tical realization of these MISO WPT systems.Table I compares the obtained capacitor values for lossless

and lossy MISO-31 WPT setups, with the receiver loop at threedifferent angles, at a distance of . As can be seen, thevalues seem not to change at all for the lossless setup (changesappear at the fourth and fifth digit behind the comma; thus, highsensitivity on actual values), since that optimized solution es-sentially remains the same, independent of the receiver position;see Fig. 7(a). The values are different overall and do changefor the lossy setup, however, particularly on the transmitter side( and ).

C. Coaxial Setups

1) Non-Planar Setups: The only geometry that never profitsfrom proximity effects is a coaxial (but not necessarily planar)arrangement of the transmitter loops, as illustrated in Fig. 11; forany elevation angle (and azimuth ), the minimum distancebetween any of the transmitters and the receiver is always thesame or more as in the SISO case.


Fig. 7. Optimized currents: MISO-31 and SISO vs. angle of elevation , at distance , for lossless and lossy loop conductors. (a) lossless, , (b) lossy,.

Fig. 8. Contours of the magnetic field (in dB) in theplane for MISO-31 setups with the receiver at 60 off broadside, .(a) Lossless loops, and (b) lossy loops, .

Fig. 9. Illustrations of the MISO-31 WPT setup and its decompositions,offset MISO-21 and offset SISO, to separate their effects on the efficiency.(a) MISO-31, (b) MISO-21 offset, and (c) SISO offset.

Fig. 10. Comparison of the separated transfer efficiency effects and the totaltransfer efficiency pattern, at distance . (a) Lossless loops, and(b) lossy loops, .

Fig. 12 shows the optimized results for a separation of, again for the distance of . As can be seen, even

in lossy cases, there is a slight increase in transfer efficiency,when using multiple loops. Clearly, since the added transmittersare further away from the receiver, this cannot be due to thesame proximity effect as before, in the planar case.However, also in this case, the working principles of the loss-

less and lossy cases are different, as Figs. 13 and 15 reveal:While in the lossless case again far-field radiation is minimizedby almost canceling currents, in the lossy case the loops are usedmore or less in parallel, thereby reducing conduction loss.2) Planar Coaxial Setup: The combination of both ideas, to

have a planar setup but multiple loops that are all at the same dis-tance (or further) from the receiver, leads to the planar coaxial


TABLE IOPTIMIZED CAPACITOR VALUES AT THE DIFFERENT ANGLES AT DISTANCE

, FOR LOSSLESS AND LOSSY MISO-31 WPT SETUPS

Fig. 11. The coaxial MISO WPT setups: multiple transmitters, with the posi-tions , where and a receiver at .(a) SISO and (b) MISO-21 coaxial, (c) MISO-31 coaxial.

(concentric) setup, known frommetasurface and subwavelengthfocusing applications [12], depicted in Fig. 14.Since, as is well known, larger transmitter loops lead to higher

flux and thus stronger coupling and higher transfer efficiency,for a fair comparison, only loops of equal or smaller radius thanthe receiver loop are considered: The loop radii arefor the MISO-21 and .The currents, capacitors, and load resistance have been op-

timized using the analytical formulations. Fig. 16 shows theresulting efficiency patterns for lossless and lossy loops. Theylook very similar to the one in Fig. 12(b) and also the workingprinciple is quite similar to the coaxial cases: While in the loss-less cases, the inner loops are used to cancel far-field effects(radiation) of the outer loop, in the lossy case all loops are usedin parallel, to minimize conduction loss. Plots of the optimal re-ceiver resistance are omitted since they look very similar toFig. 12(d).Thus, solutions with highly oscillating currents, as received

for subwavelength beamforming [12], are not feasible whentransfer (or coupling) efficiency is of the main interest.Coaxial MISO WPT setups have shown that there is poten-

tial transfer efficiency enhancement even without closer prox-imity to the receiver. However, as can be seen from Figs. 12(b)and 16(b), the potential efficiency enhancement due to multipletransmitters is smaller than the one due to impedance matching:choosing the optimal load resistance is essential to achievemaximum power transfer.

VI. SUMMARY AND CONCLUSION

The transfer efficiency enhancement potential of wirelesspower transfer systems with multiple transmitters (MISOWPT systems) was analyzed using convex optimization andLagrangian duality theory.A convex optimization algorithm (quadratic programming)

to maximize the transfer efficiency by optimizing the required

currents as well as the receiver capacitor was presented. Theexcitation voltages together with the transmitter capacitors wereobtained by least-square minimization of the amplitudes.Further, using Lagrangian duality theory, analytical solutions

of the dual optimization problem were found, with which themaximum transfer efficiency of a given arrangement and load

as well as all required parameters could be obtained, di-rectly. From these formulations the optimum load couldalso be obtained, to assess the maximum achievable transfer ef-ficiency of a given system.For problems of nonpassivity, originating from the numerical

code(s) used to obtain the (unloaded) impedance matrix of theWPT setup in the first place, a simple passivity enforcement wasapplied. The negative eigenvalues of the eigenvalue decompo-sition were replaced by positive ones, whereby both passivityand physical meaning of the model was recovered.Using these methods, performances and working principles

of planar and coaxial MISO WTP systems were analyzed andcompared for cases with lossless and lossy conductors. Losslesscases lead to solutions similar to the ones obtained for near-fieldfocusing [11], [13], with almost canceling currents to minimizefar-field radiation. This working principle is not feasible underlossy conditions, however, if (transfer) efficiency is the maininterest.Since the impedance matrix and reactances vary with fre-

quency, the optimum is only achieved at the particular frequencyof operation; how fast the transfer efficiency drops depends onthe distance, geometry and quality factor of the whole system.Over larger frequency spans, resonance splitting effects are ob-tained, similar in nature to those reported for SISO systems [14],but with more resonances (due to the higher number of res-onators involved).

APPENDIX ATREATMENT OF NON-PASSIVE MODELS

In some cases, numerical solvers such as DMBC or HFSSreturn non-passive (unloaded) impedance matrices, especiallywhen using perfect conductors or very low-loss materials. Inthese cases, the unloaded impedance matrix does not satisfy thepassivity condition [22]

(52)

which with reduces to

(53)

with the (unloaded) resistance matrix being positive definite.Due to the fact that there is at least some radiation loss, (52)and (53) actually reduce to strict inequalities.Therefore, for nonpositive semidefinite matrices, the eigen-

value decomposition returns at least one negative eigenvalue. Itis evident that the same is true for the (real) matrices and .Even if that eigenvalue is small, it will be picked up during theoptimization process by the quadratic programming algorithm(the minimization leads to ), and the solution willbe physically meaningless.


Fig. 12. Coaxial MISO-21, MISO-31 vs. SISO: maximum transfer efficiency for and versus and optimum load resistance , forand lossless, , (a), (c) and lossy, , (b), (d) loops at a distance . (a) Transfer efficiency , thick, , thin,

for , (b) Transfer efficiency , thick, , thin, for , (c) Optimum load resistance (in ), for , (d) Optimum load resistance(in ), for .

Fig. 13. Contours of the magnetic field (in dB) in theplane for MISO-31 setups with the receiver at 60 off broadside, .(a) Lossless, and (b) lossy, .

There are a variety of elaborate methods to enforce passivityavailable. However, most of them rely on the impedance matrixas a function of frequency (e.g., adequately sampled over a cer-tain frequency range) [23], [24]. While the impedance matrixcould be sampled over a certain frequency range, using elab-

Fig. 14. Planar coaxial (concentric) MISO WPT setups. (a) Planar coaxialMISO-21 and (b) planar coaxial MISO-31.

orate methods of enforcing passivity is beyond the scope ofthis discussion. As numerical studies have shown, a much moresimple method suffices in these cases.The general idea is to make the matrix positive definite by

replacing the negative eigenvalue(s) in the (diagonal) matrixobtained from the eigenvalue decomposition by a positive one,making it purely positive and resulting in a passive approx-imation:

—————- (54)

By comparing the norms of the matrices, it can be assessed howmuch of thematrix changed, and by comparing the optimal solu-tion to the corresponding eigenvector, it can be assessed of howmuch importance that particular eigenvalue/-vector actually is.The negative eigenvalue is replaced according to

(55)


Fig. 15. Optimized currents: Coaxial MISO-31 and SISO setups vs. elevation angle , at distance , for lossless and lossy conductors. (a) Lossless,and (b) lossy, .

Fig. 16. Concentric MISO-21, MISO-31 vs. SISO: maximum transfer efficiency for and versus for , lossless, , (a) andlossy, , (b) loops at a distance . (Graphs for are omitted, since they look very similar to Fig. 12(d).) (a) Transfer efficiency , thick,

, thin, for , (b) Transfer efficiency , thick, , thin, for .

where is a very small positive number, chosen smallerthan the smallest of all other eigenvalues, but large enough soas not to produce numerical problems. If the amplitude of thenegative eigenvalue is already (much) smaller than all the other(positive) eigenvalues, then could be an option, which,as tests have shown, does not lead to numerical issues.In the case of DMBC, nonpassive matrices were only ob-

tained for larger models (seven loops and more) or for very high(but finite) conductivity, e.g., . For these models, mostof the time only one, in rare cases two of the eigenvalues, turnout negative, and their amplitude is usually very small in ampli-tude, compared to all others. Thus, the replacementwas used.Since those nonpassive impedance matrices do not represent

the physics of the model anyway, it stands to reason to perturbthem just enough so as to both be able to use it for the optimiza-tion procedure as well as to get most of the physical meaningback. The graphs in the result section seem physically reason-able and serve, therefore, as evidence of successful passivityenforcement.

ACKNOWLEDGMENTThe authors would like to thank Prof. P. Triverio for helpful

discussions on the conditions for passive impedance matrices

and passivity enforcement techniques as well as pointing toSchur’s complement as means to validate the non-negative na-ture of the efficiency in (49).

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purposes (classic paper),” Proc. IEEE, vol. 87, no. 7, Jul. 1999.[2] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and

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[18] M. A. Tilston and K. G. Balmain, “On the suppression of asymmetricartifacts arising in an implementation of the thin-wire method of mo-ments,” IEEE Trans. Antennas Propag., vol. 38, no. 2, pp. 281–285,Feb. 1990.

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[20] R. C. Hansen and R. E. Collin, Small Antenna Handbook. New York,NY, USA: Wiley, Aug. 2011.

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[22] P. Triverio, S. Grivet-Tolcia, M. S. Nakhla, F. G. Canavero, and R.Achar, “Stability, causality, and passivity in electrical interconnectmodels,” IEEE Trans. Adv. Packag., vol. 30, no. 4, pp. 795–808, Nov.2007.

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Hans-Dieter Lang (S’12) was born in Switzerland,in 1984. He received the B.Sc. and M.Sc. FHO de-grees in electrical engineering from the University ofApplied Sciences Eastern Switzerland, Rapperswil,Switzerland, HSR, in 2008 and 2010, respectively.From 2010 to 2012, he was with the Institute

for Communication Systems, ICOM, Rapperswil,Switzerland, where he worked on various projectswith industrial partners, focusing on development,integration and optimization of antennas and RFsystems as well as applied research and applications

of wired and wireless communication systems. In fall 2012, he joined theElectromagnetics Group, Electrical and Computer Engineering Department,University of Toronto, Toronto, ON, Canada, where he is currently workingtowards the Ph.D. degree. His research interests include wireless powertransfer, electromagnetic optimization as well as analytical and computationalelectromagnetics.

Alon Ludwig (S’00–M’08) received the B.Sc. andPh.D. degrees in electrical engineering from theTechnion—Israel Institute of Technology, Haifa,Israel, in 2000 and 2007, respectively.From 2008 to 2010, he was a Postdoctoral As-

sociate with the School of Electrical and ComputerEngineering, Purdue University, West Lafayette,IN, USA. Currently, he is a Postdoctoral Fellowat the University of Toronto, ON, Canada. Hisresearch interests include efficient frequency- andtime-domain numerical techniques for the solution

of electromagnetic scattering problems, near-field antenna arrays for imagingand focusing applications, aspects of electromagnetic mixing formulas inmetamaterials, and bianisotropicy and chirality in metamaterials.Dr. Ludwig received a number of awards, including the Gutwirth Fellowship,

the Jury Prize, the Applied Materials Scholarship, and the Viterbi PostdoctoralFellowship.

Costas D. Sarris (SM’08) received the Ph.D. degreein electrical engineering and the M.Sc. degree in ap-plied mathematics from the University of Michigan,Ann Arbor, MI, USA, in 2002.He is currently a Full Professor and the Eugene

V. Polistuk Chair in Electromagnetic Design at theDepartment of Electrical and Computer Engineeringand an Associate Chair of the Division of Engi-neering Science, University of Toronto, Toronto,ON, Canada. His research interests are in the areaof numerical electromagnetics, with emphasis on

high-order, multi-scale computational methods, modeling under stochasticuncertainty, as well as applications of numerical methods to wireless channelmodeling, wave-propagation in complex media and meta-materials, wirelesspower transfer, and electromagnetic compatibility/interference (EMI/EMC)problems.Prof. Sarris was the recipient of the IEEE MTT-S Outstanding Young Engi-

neer Award in 2013 and an Early Researcher Award from the Ontario Ministryfor Research and Innovation in 2007. His students have received paper awardsat the 2009 IEEE MTT-S International Microwave Symposium, the 2008 Ap-plied Computational Electromagnetics Society Conference, and the 2008 and2009 IEEE International Symposia on Antennas and Propagation. He was anAssociate Editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY ANDTECHNIQUES from 2009 to 2013 and the IEEE MICROWAVE AND WIRELESSCOMPONENTS LETTERS from 2007 to 2009. He is the TPC Chair for the 2015IEEE AP-S International Sympopsium on Antennas and Propagation and theCNC/USNC Joint Meeting in Vancouver, BC, Canada.

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