IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

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Semidefinite Relaxation-Based Optimization of
Multiple-Input Wireless Power Transfer Systems

arXiv:1701.07344v1 [math.OC] 24 Jan 2017

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

Abstract—An optimization procedure for multi-transmitter
(MISO) wireless power transfer (WPT) systems based on tight
semidefinite relaxation (SDR) is presented. This method ensures
physical realizability of MISO WPT systems designed via convex
optimization — a robust, semi-analytical and intuitive route to
optimizing such systems. To that end, the nonconvex constraints
requiring that power is fed into rather than drawn from the
system via all transmitter ports are incorporated in a convex
semidefinite relaxation, which is efficiently and reliably solvable
by dedicated algorithms. A test of the solution then confirms
that this modified problem is equivalent (tight relaxation) to the
original (nonconvex) one and that the true global optimum has
been found. This is a clear advantage over global optimization
methods (e.g. genetic algorithms), where convergence to the true
global optimum cannot be ensured or tested. Discussions of
numerical results yielded by both the closed-form expressions and
the refined technique illustrate the importance and practicability
of the new method. It, is shown that this technique offers a
rigorous optimization framework for a broad range of current
and emerging WPT applications.
Index Terms—Convex Optimization, Multiple Transmitters,
Power Transfer Efficiency, Semidefinite Programming, Semidefinite Relaxation, Wireless Power Transfer.

W

I. I NTRODUCTION

IRELESS power transfer (WPT) systems have come
a long way, since their origins over a century ago [1]
and their more recent rediscovery [2]. Among many other advancements and developments, multi-transmitter WPT systems
have been investigated, both theoretically and experimentally
[3]–[6]. Leveraging the additional design parameters offered
by the use of multiple transmitters, these multiple-input singleoutput (MISO) WPT systems have been shown to outperform
their single-input single-output (SISO) counterparts. A key
aspect of the improved performance of MISO over SISO
WPT systems is the natural resilience of the former to the
effects of transmitter-receiver misalignment, which are known
to severely compromise the power transfer efficiency (PTE) of
SISO WPT systems. On the other hand, the increase in degrees
of freedom of such systems also makes it challenging to find
their optimum operating mode and estimate their maximum
achievable performance. In such cases, global optimization
methods such as Genetic Algorithms (GA) are commonly
used, see for example [7]; however, there is no guarantee that
Manuscript received September xx, 2016; revised Month xx, 2017; accepted
Month xx, 2017. Date of publication: Month zz, 2017. Current version: 00
The authors are with the Electromagnetics Group of the Department of
Electrical and Computer Engineering, University of Toronto, ON, M5S 3G4,
Canada, (e-mail: hd.lang@mail.utoronto.ca).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier XX.YYYY/TMTT.2017.abcdef

they converge to the global optimum, despite their significant
computational cost.
Recently, the fundamental physical limits of such MISO
WPT systems were determined using convex optimization [6].
The maximum achievable PTE as well as all relevant electrical
parameters (such as currents, resistive and reactive loading,
etc.) for optimal operation were derived in closed form. These
closed-form results, which are briefly reviewed in Sec. II, are
highly valuable, very intuitive and physically meaningful.
However, it still possible that insufficiently constrained
optimization of WPT systems may lead to solutions where
power is drawn from the system via some transmitter ports,
instead of being fed into it. Such solutions are mathematically
valid optima, whose performance (high PTE) is owed to the
“recycling” of power among transmit ports suppressing the
total transmit power. Yet, they are undesirable and impractical
from a physical point of view.
Hence, the formulation of alternative techniques for the optimization of MISO WPT systems is strongly motivated, where
such operating conditions are avoided from the beginning.
As will be discussed in detail, adding such transmit power
constraints to the original optimization problem is not trivial,
as they are nonconvex; thereby breaking convexity of the entire
optimization problem. As a result, the asssociated optimization
problem is unsolvable in general, as the computational cost
grows rapidly when increasing the number of transmitters.
In the following, a MISO WPT optimization strategy based
on tight semidefinite relaxation is presented, whereby the
nonconvex problem is reformulated into a convex form with
slightly looser constraints, which can be efficiently and reliably
solved by dedicated algorithms. A simple test of the solution
then proves that this relaxation is tight, i.e. fully equivalent
to the original problem and that the true global optimum was
found. With this framework, these nonconvex transmit power
constraints can be efficiently included in the optimization process, ensuring the practical realizability of the proposed MISO
WPT systems. Notably, this new optimization framework is far
more general, powerful and versatile than the one previously
provided in closed form [6], while retaining its mathematical
rigor.
This paper begins with a brief introduction and a review
of the closed-form expressions [6]. Then, it is shown in
detail how the additional constraints are incorporated into
the optimization procedure, how its semidefinite relaxation
is derived and how the optimization result can be tested
for tightness/equivalence to the original nonconvex problem.
Finally, simulation results demonstrate the practicality of the
proposed method and the importance of including nonconvex

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transmit power constraints to obtain practically realizable
MISO WPT systems.
Remarks on the notation: Italicized thin letters represent
scalars, bold small letters refer to vectors, and bold capital
letters are matrices; vT stands for the transpose of v, while
vH stands for its Hermitian (conjugate transpose). Real and
imaginary parts of complex quantities are marked by (·)0 and
(·)00 , respectively, i.e. α = α0 + jα00 . The complex conjugate
is denoted by a bar (overline). The symbols  () and ≺
() are used to denominate positive (semi-) definiteness and
negative (semi-) definiteness of matrices, respectively. For
vectors, they stand for elementwise positive (nonnegative) and
negative (nonpositive) entries. A star (·)? marks the optimized
arguments that lead to the optimal solution of an optimization
problem.
Remarks on the terminology: ‘Program’ is a synonym for
‘optimization problem’, commonly used in the context of
mathematical optimization [8]. Further, in this paper, ‘feasibility’ will refer to satisfying constraints within a program;
i.e. a feasible solution satisfies all required constraints.
II. P RELIMINARIES
A. Power Transfer Efficiency (PTE)
The central figure of merit when optimizing WPT systems
is the power transfer efficiency (PTE) [3]–[6], [9], [10]: the
ratio of the power PL transferred to the load RL to the total
transmit power provided by the source Pt
PL
RL
PL
=
=
.
η=
Pt
Pl + PL
Rl + RL

(1)

Pl is the total power absorbed by the system, due to dissipation
and radiation, modeled by the loss resistance Rl , as illustrated
in Fig. 1.
Pl
Rl

B. The MISO WPT System Model
Let the (unloaded) impedance matrix Z of the MISO WPT
system be partitioned according to




Z=



Zt

zTtr

ztr 

 ∈ CN ×N ,


zr

where the subscripts t and r refer to the transmitter and
receiver parts, respectively, and tr stands for the elements
coupling the former to the latter.
The diagonal of Zt and zr refer to the loss resistances and
self-reactances of the transmitters and receiver, respectively.
00
Typically, zn,n
, zr00 > 0 (inductive) when considering systems
of magnetically coupled loops or coils. Likewise, the offdiagonal entries of Zt and the coupling vectors ztr contain the
mutual inductances jωMn,m . Generally, each Mn,m is complex, due to retardation effects when the electrical distances
between the transmitter(s) and/or receiver are not very small.
For physical reasons, impedance matrices have to be
positive-real [11]: The real parts of all matrices and submatrices have to be positive-definite, i.e. Z0 , Z0t  0 and zr0 > 0.
Positive definiteness of a matrix means that the quadratic form
of such a matrix is always positive:
iH Z0 i > 0

∀ i ∈ CN

η

Transmitter(s)

Power Transfer System

RL

PL

Receiver (load)

Fig. 1. Illustration of general power transfer systems and the PTE as defined
in (1): The ratio of the power transferred to the load, PL , and the total
transmitted power, Pt . Loss (e.g. due to conduction losses and radiation) is
modeled by the resistance Rl .

Note that, within the context of this paper, PTE refers only
to the electromagnetic power transfer efficiency of the system.
Any practical realization of a WPT system will have an end-toend efficiency which is limited by this PTE; signal generation
and impedance matching on the transmitter side as well as
matching and rectification will lower the overall performance,
but are not considered here. Further, RL is only practically
realizable from a certain resistance level on. However, in this
paper the maximum achievable performance in view of the
electromagnetic PTE is of central interest, which includes the
question of the optimal load resistance.

(3)

This mathematical property represents the fact that there is
always some amount of loss in the system. Positive semidefiniteness ( 0) implies nonnegative (≥ 0) quadratic forms
and for negative (semi-) definiteness the opposite signs and
directions apply.
The process of adding reactances to each of the transmitter
and receiver nodes as well as a resistive load to the receiver
will be referred to as loading of the WPT system, where Ẑ is
the resulting loaded impedance matrix:
Ẑ = Z + jX + RL .

Pt

(2)

(4)

The real-valued diagonal matrix X contains the load reactances (positive and negative values referring to inductive
and capacitive elements, respectively). The matrix RL is zero
everywhere but at the last diagonal entry, corresponding to the
receiver, where the load resistance RL > 0 is located.
In detail, the voltages and currents v, i of the whole (loaded)
WPT system are related according to
Ẑ
}|
{



 
! 


Zt ztr
Xt 0
0 0
it
vt
=
+j
+
0 xr
0 RL
ir
vr = 0
zTtr zr
| {z } | {z } | {z } | {z }
| {z }
v
Z
X
RL
i
(5)
where, similarly to the impedance matrices, the subscript t and
r refer to the transmitter and receiver voltages and currents,
respectively. Since the load resistance RL is actually part of the
impedance matrix, KVL states that the corresponding receiver
voltage is zero, i.e. vr = 0, as illustrated in Fig. 2.
z

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and the resulting PTE is given by

Loaded WPTS: Ẑ = Z + jX + RL
Unloaded WPTS: Z
TX
RX
zr
Zt

C1

i1
v1

ηmax =
Cr
ir
vr = 0

ztr
RL

CN −1

C2

iN −1

i2
v2

vN −1

Fig. 2. Loop-based MISO WPT system model: Core structure with unloaded
impedance matrix Z (obtained via simulation or measurements), reactive
components, here xn = −(ωCn )−1 , resistive load RL and voltage sources
vn added to the loaded impedance matrix Ẑ. Note that at the receiver end,
since RL is part of Ẑ, the voltage at the corresponding port is zero, vr = 0.

To maximize the performance of such systems, the aim is
to find optimal voltages v and currents i (real and
T
 imaginary
=
parts) as well as loading components x = xTt , xr
diag(X), RL , which maximize the PTE, obtained according
to its definition (1) as
η=

1 H
2 i RL i
1 H
0
2 (i v)

=

iH RL i
.
iH (Z0 + RL )i

(6)

As set out above, the PTE can be given in terms of the currents
i alone. The voltages v follow from (5), but are only physically
meaningful as long as vr = 0 is ensured.
C. Review of the Convex Optimization of MISO WPT Systems
As pointed out previously [6], the biquadratic fraction (6)
with both convex numerator and denominator, does not have
to be convex itself. However, convexity is the property that
ensures that the objective has a single and global optimum
that can be reliably found by dedicated algorithms [8].
Note that there are numerical algorithms dedicated to dealing with this type of problem directly, often referring to it
as maximization of the constrained Rayleigh quotient [12],
[13]. However, in the following, a more direct and physically
intuitive way of dealing with this problem is taken.
In order to obtain a uniquely defined optimization problem,
the receiver current could be assumed to be a known constant
ir ∈ R, ir 6= 0, (purely real, non-zero). More specifically
and without p
loss in generality, the receiver current is chosen
2/RL , leading to unit transferred power, i.e.
to be ir =
PL = 12 iH RL i = 12 RL i2r = 1. Then, maximizing the PTE (6)
is equivalent to minimizing the power loss
Pl =

1 H 0
i Zi
2

1  0T 0 0
0 00
0T 0
2 0
=
it Zt it + i00T
. (7)
t Zt it + 2ir ztr it + ir zr
2

1
.
Pl,min + 1

(8)

In order to only have to deal with real variables in the following steps, the complex transmitter currents it are separated
into real and imaginary parts i0t and i00t , respectively:
 0
i
(9)
ct = 00t .
it
To satisfy KVL at the receiver end, i.e. ensuring vr = 0 as
in (5) and illustrated in Fig. 2, the following linear constraint
(affine [8] in ct ) has to hold:
r
 
 0T

2 ẑr0
ztr −z00trT
= 0,
(10)
ct +
z00trT
z0trT
RL ẑr00
where ẑr = zr + RL + jxr is the loaded receiver selfimpedance. Since the receiver reactance xr is a free parameter,
it can always be chosen so as to automatically satisfy the
second constraint for the optimal transmitter currents c?t ,
r
RL  00 T 0 T  ?
?
00
ztr ztr ct .
xr = −zr −
(11)
2

Thus, the second row of the constraints (10) can be removed.
Using the objective (7) in terms of the real currents (9) in
combination with the remaining constraint of (10), the program
to solve for optimal wireless power transfer to a specific load
RL at the receiver can be given in detail as
r
 0

1
2  0T 
zr
Zt 0
z
0 ct +
Pl,min = min cTt
c
+
t
0
0 Zt
ct 2
RL tr
RL
r
 0 T 00 T 
2 0
z
z
s.t. tr tr ct = −
(z + RL )
RL r
(12)
The final optimization problem (12) constitutes a convex
quadratic program (QP) in terms of the transmitter currents
ct . Such programs can efficiently and reliably be solved to
(nearly) arbitrary precision using numerical methods, such as
interior-point or active-set algorithms [8]. More importantly
however, the fact that (12) is a purely equality-constrained
convex QP means that it has an analytic solution, obtainable
for example via Lagrangian duality [8]. The steps involved
are straightforward and, therefore, omitted here. These closedform expressions can then be optimized with respect to RL , by
setting its derivative to zero. In the following, the so obtained
closed-form expressions of [6] are reviewed and presented in
a more physically meaningful and intuitive form.
D. Closed-Form Expressions for Optimal MISO WPT Systems
1) Fundamental Quantities: The optimal operating parameters, such as transmitter currents and loading elements, as well
as the resulting PTE are found in terms of two fundamental
quantities with physical interpretation:
• Minimum-loss output impedance
0
zo = zo0 + jzo00 = zr − zTtr Z0−1
t ztr ∈ C

(13)

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Mutual coupling quality factor (also called “mutual Q”)
s
0−1
zH
tr Zt ztr
U=
∈R
(14)
zo0

Since all impedance (sub-) matrices are positive-real, it follows
that zo0 , U > 0.
The real part of the minimum-loss output impedance zo
corresponds to the (minimized) loss due to radiation and
dissipation (i.e. Rl = zo0 in (1) and Fig. 1), as illustrated
in Fig. 3, whereas the imaginary part zo00 is the reactive
component to be canceled out by the receiver reactance xr .

v1
Pt
vN −1

i1

Pl,min

x1

zo
Z

iN −1 xN −1

xr ir
RL

the maximum achievable PTE (the physical limit of the WPT
system) results:
ηmax = ηres |RL =R? =
L

U2
√
.
(1 + 1 + U 2 )2

(19)

?
Note that ηmax and RL
are of the same form as for SISO
systems [9], [14], where U is given by (15). In other words,
SISO WPT systems are special cases of the more general
MISO WPT systems, since in the case of a single transmitter,
(14) collapses to (15) and real part of (13) is replaced by Rt ,
as previously mentioned; the rest of the expressions remain
the same.

III. P ROBLEM S TATEMENT

PL

A. Port Impedance Matrices (PIMs) Tn
The total transmit power in the denominator of (1) can be
separated into the contributions of each port n:

Ẑ
Fig. 3. The optimized MISO WPT system, including the minimal output
impedance zo responsible for both losses during the power transfer as well
as output reactance to be compensated.

The presented mutual coupling quality factor is the natural
extension of the well-known “mutual Q” for SISO systems
[9], [14], commonly given as
U =k

p
ωM
Qt Qr = √
Rt Rr

(15)

Note that generally the mutual inductance M and also the
coupling coefficient k are complex quantities. In such cases,
their absolute values should be used in (15). Evidently, the
real part of the output impedance zo0 corresponds to the loss
resistance on the receiver side Rr , whereas dividing by the
loss resistance on the transmitter side Rt is replaced by the
inverse of the real part of the transmitter impedance matrix
the mutual
Z0t . Thus, in terms of quality and coupling factors,
q
coupling quality factor (14) becomes U = kH Qt kQr for
MISO systems.
2) Optimal Currents, Loads and PTE: The optimal transmitter currents are found as


zo + RL + jxr
0
i?t = −Z0−1
z
(16)
z
+
tr ir .
t
tr
zo0 U 2
The optimal receiver reactance is canceling the reactive
component of the output impedance, as mentioned, x?r = −zo00 .
The transmitter reactances xt = diag(Xt ) are not essential
for maximizing the PTE. They can be chosen arbitrarily, as
long as the optimal transmitter currents result from applying
the voltages to the loaded system.
Further, using the optimal receiver reactance x?r , the system
operates at the resonant PTE, given by
ηres = η|xr =x?r =

Finally, when in addition to the reactance also using the
optimal load resistance
p
?
RL
= zo0 1 + U 2
(18)

U2
RL
·
.
0
2
1 + RL /zo + U
RL + zo0

(17)

1
1
Pt = Pl + PL = (iH v)0 = iH Ẑ0 i
2
2
X
1
1X H
=
Pt,n = (vn in )0 =
i T̂n i
2
2 n
n

(20)

where T̂n are going to be referred to as port impedance
matrices (PIMs). These PIMs are interesting in many ways:
• They sum up to the total loss resistance matrices:
X
X
Tn = Z0 and
T̂n = Ẑ0 .
(21)
n

•

•

n

Note that Tn = T̂n , for all n except n = N , where
T̂N = TN + RL .
While Z0 and Ẑ0 are purely real-valued, symmetric
and positive definite as previously noted, the PIMs are
complex-valued, Hermitian, singular and indefinite.
Each Tn , T̂n has exactly one positive, one negative
and N − 2 zero eigenvalues. These eigenvalues and the
corresponding eigenvectors can be derived analytically,
as laid out in appendix A.

B. Nonconvex Transmit Power Constraints
The total transmit power is always Pt > 0 (strictly positive), when transferring power PL > 0. However, this is
not necessarily true for the transmitter powers at each port
n = 1, ... , N − 1, since all Tn are indefinite, as previously
mentioned.
As will be shown in the results section, in some cases, the
optimal currents (16) lead to one or more transmitter powers
being negative. This implies that via some transmitter ports,
power is drawn from the system, rather than fed into it. Such
negative transmitter powers lead to a reduced net transmit
power (20), which has a positive effect on the PTE (1).
While mathematically correct and physically meaningful
overall (at least from a circuit theory point of view), such

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solutions prove difficult to realize in practice, as discussed in
more detail in Sec. V. Therefore, it makes sense to try to avoid
such solutions altogether, during the optimization process.
In order to accomplish that, the constraints
Pt,n =

1 H
i Tn i ≥ 0
2

n = 1, ... , N − 1

(22)

would have to be added to (12). They ensure nonnegative
transmit powers and, thus, power being fed into the system
at all transmitter ports, rather than being drawn from it.
Adding the transmitter power constraints (22), the program
(12) in terms of complex currents i becomes:
1 H 0
i Zi
2
1
s.t. iH Tn i ≥ 0
n = 1 , ... , N − 1
2
 T

vr = ztr , ẑr i = 0
r


2
ir = 0, 1 i =
RL

Pl,min = min
i

(23)

Unfortunately, since Tn  0 (indefinite and not negativesemidefinite, as required to make (22) convex), the added
constraints are nonconvex, rendering the whole optimization
problem nonconvex. Thus, (23) is a nonconvex QCQP [8]
(as opposed to the convex QCQP mentioned at the end of
Sec. II-C). Note that the same would be true for constraints
of the form 12 iH Tn i ≤ Pt,n,max , i.e. limiting the maximum
transmit power at each port.
Nonconvex QCQPs belong to the class of NP-hard problems [15]. This means that there are no known algorithms
which can solve the problem for an arbitrary number of
transmitters.
IV. S EMIDEFINITE R ELAXATION
In the following, a method is presented which incorporates
the nonconvex transmit power constraints while remaining
computationally efficiently and reliably solvable.
A. Nonconvex QCQP
Zero KVL in the imaginary part can always be achieved
by appropriately choosing the receiver reactance xr and the
imaginary part of the receiver current is therefore chosen to
always remain zero. Thus, the actual problem size is M =
2N − 1 and the unknown currents are (in real form)
 0
it
c = i0r  ∈ RM .
(24)
i00t
The nonconvex QCQP (23) can be formulated in standard
QCQP form as follows:
Pl,min = min cT Q0 c
c

s.t. cT Qn c ≥ 0
Ac = b

n = 1 , ... , N − 1

(25)

where the objective and the inequality constraints contain the
leading principal M × M submatrices of the real block matrix
representations:


1 T0n −T00n
(26a)
Qn =
00
0
2 Tn Tn M ×M
P
and Q0 = n Qn .
Similar to the real part of the impedance matrix and the
PIMs, Q0  0 while Qn  0. The equality constraints
represent the KVL at the receiver node and the fixed receiver
currents (real and imaginary parts). The affine equality constraints Ac = b are actually

 0


0
ztr zr + RL −z00tr 0
p
(27)
c=
0
1
0
0
2/RL

to ensure KVL at the receiver end (vr = 0) as well as unit
transferred power PL = 1, as before.
Using the cyclic property of the trace cT Qi c =
tr(cT Qi c) = tr(Qi ccT ) the objective and all inequality
constraints can be written in linear terms of the (quadratic)
current matrix C = ccT  0:
Pl,min = min tr(Q0 C)
C,c

s.t. tr(Qn C) ≥ 0

n = 1 , ... , N − 1

(28)

Ac = b
C = ccT
Note that the three programs (23), (25) and (28) are mathematically fully equivalent. The non-convexity of the former
two has been isolated in the last condition of (28), since the
traces are linear operations in terms of the matrices C.
B. Semidefinite Relaxation in Partially Conic Form
The idea is to exchange the nonconvex constraint with a
practically equivalent, yet convex one. Similar to x = y being
equivalent to both x ≥ y and x ≤ y, C = ccT is equivalent
to both C  ccT and C  ccT . The semidefinite relaxation
(SDR) arises from removing the second inequality constraint
and using
C − ccT = 0

SDR

−−−→

C − ccT  0

(29)

Implicitly, this also ensures C  0, since ccT  0.
Via Schur complement, the relaxed constraint (29) is equivalent to


C c
C − ccT  0 ⇔
0
(30)
cT 1
which is the most common way to express the semidefinite
(inequality) constraint. Note that another formulation of the
last constraint in (28) would be to require that the rank of
the composite matrix in (30) be 1. This implicitly ensures
both symmetry and C = ccT , while the equality constraint
Ac = b ensures a nonzero result.

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Thus, the relaxed semidefinite program (SDP) of (28), using
the semidefinite constraint (30) is
relax
Pl,min
= min tr(Q0 C)
C,c

s.t. tr(Qn C) ≥ 0

Ac = b


C c
0.
cT 1

n = 1 , ... , N − 1

(31)

and represents the dual of the well-known Shor relaxation [15]
with added affine equality constraints. This problem can be
implemented readily in Matlab using CVX [16], [17], a
package for specifying and solving convex programs such as
this SDP. The actual numerical algorithm used is SDPT3 [18],
[19], which for example for a MISO-3 system (problem size
M = 7) usually converges within ten to thirty iterations, in
just a few seconds.
With this relaxed program, a lower bound on the minimum
power loss and, thus, by application of (8), an upper bound on
relax
relax
the PTE is found: Pl,min
≤ Pl,min , corresponding to ηmax
≥
ηmax . If equality is achieved, the relaxation is referred to
as tight [20], referring to the fact that the bounds have zero
gap. Evidently, then (31) is a fully equivalent reformulation of
(23), (25) and (28) and solving the (convex) SDP is equivalent
to solving the original (nonconvex) QCQP. In essence, this
also implies that the removed constraint C  ccT is naturally
satisfied by the rest of the program.
Note that, in general, relaxations where the constraints were
relaxed (but not the objective) necessarily lead to a remaining
feasibility problem, unless they are tight. By removing a
constraint, the feasible region for solutions was enlarged and,
therefore, the so found solution cannot be feasible for the original problem unless that constraint was loose, i.e. otherwise
satisfied to begin with. The remaining feasibility problem is
then to find “the next best” solution which is feasible to the
original optimization problem but has the smallest gap to the
relaxed optimum. Such problems can generally be arbitrarily
difficult to solve, as they essentially contain the original nonconvexity and therefore must still be NP-hard.
In this case, a good candidate of a feasible solution is found
directly, since the vector c? (optimal solution to c) necessarily satisfies the affine equality constraints (representing the
KVL at the receiver end). However, that vector only satisfies
the other constraints if the relaxation is tight. Methods to
find bounds on these types of inequalities are available, see
e.g. [21], but deemed not strict enough to be useful in this
case.
As it turns out, while it is intuitive and numerically convenient, the SDP form with a separate vector for the affine
constraints (31) is mathematically difficult to deal with, when
attempting to prove tightness.

C. Semidefinite Relaxation in Purely Conic (Quadratic) Form
1) Quadratic Forms of Affine Constraints: Instead of defining the receiver current via affine equality constraints, the

received power is fixed in its quadratic form directly, i.e.
1 2
0
00
2 ir RL = 1 (since ir = ir and ir = 0). Let


1 RL 0
R=
(32)
2 0 RL M ×M

be the leading principal submatrix of the block form of RL ,
similar to (26). Due to the row/column reduction, R  0
has only one non-zero entry, the N th entry on the diagonal,
where RL /2 ≥ 0 resides. Then, the condition that ensures
unit transferred power (and thereby fixes the real part of the
receiver current) becomes:
1 2
i RL = 1 ⇔ tr(RC) = 1 .
(33)
2 r
Further, since the only remaining constraint is homogeneous,
it can also be represented
in quadratic form, without loss of

generality. Let k = z0tr , zr + RL , −z00tr and K0 = kkT , then
kT c = 0

⇔

tr(K0 C) = 0

(34)

is responsible to satisfy KVL at the receiver end, i.e. to ensure
vr = 0.
For increased numerical accuracy, the following redundant
matrix equalities may be added
tr(Km C) = 0
T

m = 1, ... , M

(35)

kuTm ,

where um , is a unit column
where Km = um k +
vector of zeros everywhere but at the mth position. Note that,
while K0  0, Km  0 for all m = 1, ... , M .
Thus, incorporating the constraints (33) and (34), the
semidefinite relaxation of (25) in purely quadratic (conic) form
is obtained as:
relax
Pl,min
= min tr(Q0 C)
C

s.t. tr(Qn C) ≥ 0

n = 1 , ... , N − 1

tr(Km C) = 0

m = 0 , ... , M

(36)

tr(RC) = 1
C0

Note that the programs (31) and (36) are fully equivalent.
2) Duality: The Lagrangian of the semidefinite program
(36) is obtained by adding the constraints, weighted by penalty
factors, called dual variables [8], to the objective:
"
# !
N
−1
M
X
X
L = tr
Q0 −
λn Qn −
νm Km − σR C + σ
n=1

m=0



= tr [P(λ, ν) − σR] C + σ

= tr QC + σ

(37a)

(37b)
(37c)

where all λn ≥ 0 (and therefore the vector λ  0), in order to
only punish tr(Qn C) < 0. Note that P = PT and thus also
Q = QT .
The original problem, when considering duality called the
primal problem, can be obtained from the Lagrangian by
minimizing the objective given that an inner maximization
ensures that all constraints are satisfied:
relax
Pl,min
= min max L ,
C0 λ0
ν,σ

(38)

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

7

If one or more of the constraints are not satisfied, the Lagrangian is unbounded and diverges to +∞. Thus, (38) and
(36) are equivalent.

are
C?  0

tr(RC ) = 1 Primal equality constraint M + 1
tr(Qn C? ) ≥ 0

(39)

λ0 C0
ν,σ

As can easily be seen from (37c), the dual problem is only
bounded as long as Q is positive semidefinite:
(
0
Q0
−∞ otherwise

(40)

Thus, the dual problem really is
dual
relax
Pl,min
= Pl,min
= max {σ : Q  0}

(41)

λ0
ν,σ

(42c)

Primal ineq. constr. 1, ... , N − 1

(42d)

Dual feasibility

(42e)

Dual ineq. (semidef.) constraint

(42f)

tr(Q C ) = 0 Complementary slackness (CS)

(42g)

λ?  0
?

where the right hand side is called the dual problem. Since
the original problem is convex and optimal feasibility can be
presumed for physical reasons, strong duality [8] is expected
to hold, leading to equality in (39).

C0

(42b)

?

min max L ≥ max min L .


min tr QC =

(42a)

tr(Km C ) = 0 Primal eq. constraints 1, ... , M

General (weak) duality [8] states that

C0 λ0
ν,σ

Primal feasibility

?

?

Q 0
?

?

If the relaxation is tight, rank C = 1 or in other words
C has exactly one non-zero eigenvalue and c? is the corresponding eigenvector, i.e. C? = c? (c? )T . Thus, to test
tightness, either the eigenvalues of C? can be investigated,
or, numerically more efficient, the normalized tightness error,
defined as
C? − c? (c? )T
=
(43)
(c? )T c?
can be used. As long as  is small (within the bounds of
typical numerical approximation errors), this proves that the
semidefinite relaxation is tight and, therefore, provides an
exact solution c? to the nonconvex QCQP (28). Numerical
experiments showed that this seems always the case (down to
typical normalized tightness errors around  ≈ 10−12 ) and that
C? has only one nonzero (and indeed positive) eigenvalue and
c? is its corresponding eigenvector.

which is again an SDP, as required (since SDPs are self-dual).

D. Optimizing the Receiver Load RL

3) Test of Tightness: The Karush-Kuhn-Tucker (KKT) conditions [8] of optimality for the two problems (36) and (41)

The presented method addresses optimization of the transmitter currents and receiver capacitance, but leaves the receiver
z

z

3=0
30

planar MISO

3=0

θ

!30

30

y

θ

!30

x

y

x

60

60

!60

!60

z

3=0
30

θ

!30

y

90
Pt = !20 !10

x

0

10

!90
20 W

90
Pt = !20 !10

(b)

60

0

10

!90
20 W

(c)

!60
z

3=0
90
Pt = !20 !10

0

10

!90
20 W

30

z

3=0

θ

!30

30

y

x

(a)
60

θ

!30

y

x

60

!60

!60

coaxial MISO
90
Pt = !200 !100

(d)

0

!90
100 200 W

90
Pt = !200 !100

0

!90
100 200 W

(e)

Fig. 7. Comparison of the transmit power patterns of WPT systems optimized via closed-form expressions (13) to (19), [6]: (a) SISO reference, (b) and (c)
planar configurations MISO-2p and -3p, respectively, as well as coaxial setups MISO-2c and -3c, (d) and (e), respectively. Negative transmit power mean that
power is drawn from, instead of fed into, the system via that particular transmitter port, referenced via the colors. Distance of the receiver d = λ/10. Note
the different scaling of (d) and (e) with respect to all the others.

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8

load resistance RL unchanged. RL could be optimized numerically in an outer loop (while optimizing the system for each
RL in the inner loop); as the optimal PTE is a concave function
in RL . However, it has been observed that the maximum PTE
is usually not sensitive to the load resistance RL , but strongly
depends on xr ≈ x?r [22]. Thus, even using the closed-form
expression (18), results that are very close to the absolute
optimum can be expected, as long as xr is optimized.
V. E XAMPLES
Fig. 4 shows the geometrical setups of the basic SISO and
MISO WPT systems under consideration. The frequency of
operation is f = 40 MHz and the coils are single-turn loops
with radii of rloop = λ/100, where λ is the freespace wavelength at the operating frequency. The conductors are made of
copper (σ = 5.8×107 S/m) wire with a thickness (wire radius)
of rwire = rloop /10. The spatial separations of the multiple
transmitter loops are ∆x = λ/50 and ∆z = λ/100, in the xand z-directions, respectively. The receiver loop position is in
the xz-plane (y = 0), at a distance d = 0.05λ, ... , 0.2λ from
the center of the transmitter (array), as specified for each of
the following results, and at the angle θ off broadside (off the
z-axis). Note that all the loops are parallel to the xy-plane.
z

z

z

z

z

RX

θ

θ
y

TX

y
x

x
(a)

θ

θ

x

x
(c)

y

y

y
x

(b)

θ

(d)

(e)

Fig. 4. Illustrations of the SISO WPT reference (a) and the MISO WPT
configurations under consideration: Two planar configurations MISO-2p (b)
and MISO-3p (c), with transmitters located at the positions (xi , yi , 0), and
two coaxial configurations MISO-2c (d) and MISO-3c (e) with transmitters
at (0, 0, zi ). In all cases a single receiver loop is located at (d, θ, φ = 0).

For all the results in this paper, the unloaded impedance
matrices were obtained via full-wave simulation using the
Multiradius Bridge-Current (MBC) method, a computationally
efficient and accurate wire-based method of moments (MoM)
code with sinusoidal current elements [23], [24].

setups also provide enhanced performance, predominantly in
the broadside direction. In these cases, the transmitters are
often being used “in parallel” (as observed when looking
at the currents), so as to minimize conduction losses. The
performance enhancement appears to become less significant
when further increasing the number of transmitters.
2) Optimal receiver capacitance: Fig. 5(b) shows the optimal receiver capacitances obtained via the closed-form expressions. It is observed that the capacitance values remain
fairly close to about 68.86 pF, over all angles.
As mentioned in the introduction of Sec. IV, previous
studies [22] revealed that achieving the maximum possible
PTE is generally very sensitive (particularly for such singleturn loop-based MISO setups) to the receiver capacitance
values; commonly tolerances much lower than one percent
are obtained, to maintain a performance within 1 % of the
optimum PTE. On the other hand, the sensitivity on the load
resistance RL is much smaller; often tolerances of 10% and
higher are obtained for the same permissible decrease in
performance.
3) Positive and Negative Transmit Powers: Fig. 7 shows the
underlying working principles of these MISO WPT systems
in greater detail, considering the individual transmit powers
Pt,n as defined in (22): The first graph (a) shows the transmit
power of the SISO reference system as a radial function of
the receiver position angle θ. As is well known, there are two
ranges of angles (around approx. ±60◦ off broadside) at which
the transmitter loop couples very weakly to the receiver loop
(in fact, infinitely small loops would have points where they
do not couple at all). In those areas, the transmitter power
spikes to very high values, in order to still transfer unit power
(PL = 1 W) to the receiver.
3=0
30

!30

60

!60

90
2max = 0

0.4

0.6

!90
0.8

(a)

A. Closed-Form Optimized MISO WPT Systems
1) Maximum achievable PTE: As expected, MISO WPT
systems can provide superior performance compared to the
SISO reference system: Fig. 5(a) shows typical PTE patterns
as a function of the receiver position angle θ at the distance
d = λ/10 to the transmitter center. Two main observations
can be made: First, all the MISO setups outperform the SISO
system at every angle θ. However, overall, the increase in
PTE is more significant for the planar MISO systems, than
for the coaxial ones. This is due to the fact that at angles
off broadside (θ 6= 0), the distance to one of the transmitters
is always smaller than d. Thus, it has to be expected that
the transmitter closest to the receiver is favored over the
others and that all (or at least most) the power is transferred
to the receiver via that particular one. Second, the coaxial

0.2

3=0
30

!30

60

90

!60

!90
Cr = 68.82 68.84 68.86 68.88 pF
(b)

Fig. 5. Comparison of the patterns of the maximum achievable PTE ηmax (a)
and receiver capacitance Cr? (b) obtained via closed-form optimization (13)
to (19), for the planar and coaxial MISO WPT configurations Fig. 4 (using
the same color code) as well as the SISO reference, at the distance d = λ/10.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

z

9

3=0

3=0

30

30

!30

!30

θ

3=0

0.056
0.106
0.156

30

0.056
0.106
0.156

!30

y

60

x

!60

90
Pt = !10

!5

0

5

60

!60

!90 90
10 W

"2max = !50

(a)
z

!40

!30

60

!90 90
!20 dB

!60

106 #"Cr /Cr = !60 !30

(b)

3=0
30

!30

30

!90
60

(c)

3=0

30

0

!30

θ

3=0

0.056
0.106
0.156

30

0.056
0.106
0.156

!30

y
x

60

!60

90
Pt = !10

!5

0

5

!90 90
10 W

60

!60

"2max = !50

(d)

!40

!30

!90 90
!20 dB

60

!60

106 #"Cr /Cr = !200 !100

(e)

0

!90
100 200

(f)

Fig. 9. Transmit powers Pt,n of closed-form optima (thin, dashed) and SDR results (thick, solid) at distance d = 0.1λ (a, d), PTE reduction with respect to
closed-form optimization ∆ηmax in dB (b, e), and normalized deviation of receiver capacitance ∆Cr /Cr (c, f) of the planar multi-transmitter configurations
MISO-2p (a-c) and MISO-3p (d-f).

The top right two graphs, in Figs. 7(b) and (c), show the
transmit power patterns of the planar MISO setups, whereas
(d) and (e) on the bottom right show the coaxial multitransmitter cases. In the planar cases, most of the power is
always transferred via the closest transmitter, as expected. In
addition, the remaining transmitters appear to draw away a
part of the remaining power in other directions, and feed it
back into the system. A similar observation can be made for
the coaxial setups: In the regions around the ±60◦ angles,
most of the power is inserted into the system by the first
(closest) loop, while the others are used to feed some power
back into the system. However, at angles close to broadside,
most of the power is transferred via the farthest loop, with the
remaining loops acting as directors (and also feeding power
back into the system). Depending on the distance d between
the transmitter center and the receiver, the amplitudes of these
negative transmit powers can be quite significant. Note that
all loops are modeled with conduction loss and, therefore,
extracting power also comes at a cost, unlike in lossless cases;
this is already accounted for during the optimization.
B. MISO WPT Systems with Transmit Power Constraints
Fig. 8 compares one example (MISO-3c, with the color
scheme in accordance with the setup shown in Fig. 7(e), at
θ = 18◦ off broadside and d = λ/10) of an optimal solution
from the closed-form expressions that involves negative transmit powers (a) and compares it to its nonnegative counterpart
(b), to shed light on the difference between the underlying
mechanisms. In the first case, power is fed into the system
via two transmitter ports, while one transmitter port extracts
power. In turn, the net power supplied to the system, given by
the sum of all transmitter port powers, is much smaller than the
maximum transmit power (but due to the positive-definiteness

Pl
Pt,1 = −12.07 W
Pt,2 = 2.94 W

η

PL = 1 W

RL

Pt,3 = 11.35 W
MISO-3c WPT System

Pt = 2.22 W
(a)

Pl
Pt,1 = 0 W
Pt,2 = 0 W

η

PL = 1 W

RL

Pt,3 = 2.22 W
MISO-3c WPT System

Pt = 2.22 W
(b)
Fig. 8. Examples with and without negative transmit powers: Similar
performance (max. PTE of ηmax = 1/2.22 ≈ 45% in both cases), but
case (a) involves harvesting power from one port, while (b) involves only
nonnegative transmit powers; the loops connected to the first two ports are
only used passively.

of the impedance matrix (3) always remains positive). It is
implicitly assumed that the power extracted from the system
is fully harvested and can be completely “reused”, i.e. fed

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back into the system at 100% efficiency. Evidently, this poses
difficulties when trying to realize such a system in practice, as
the power cannot be easily “recycled” — at least not at perfect
efficiency.
The nonnegative transmit power solution in Fig. 8(b)
achieves approximately the same performance by only inserting power via the last port and using the optimally tuned
transmitter antennas connected to the other two ports passively.
This operating mode can be implemented in practice in a
straightforward fashion. Additionally, with all transmit powers
being positive, they also become smaller in amplitude, which
further simplifies the realization of such systems.
Figs. 9 and 11 compare transmit powers, performance degradation and deviation in receiver capacitances in polar form
for the two planar and the two coaxial MISO WPT systems,
respectively. Fig. 10 shows the corresponding normalized
tightness errors, confirming that the relaxation is tight and the
solutions solve the original (nonconvex) problem.
The transmit power patterns in the graphs Figs. 9 and
11 (a) and (d) demonstrate that the semidefinite relaxation
does indeed lead to solutions where no power is drawn from
any transmitter port, as can be seen by comparing the new
results (solid thick lines) to the closed-form solutions (thin
dashed lines). Note that the optimal constrained transmit power
solutions are different from simply setting formerly negative
transmit powers to zero (this in fact would both lead to
inferior performance as well as break the normalization to
unit received power). The performance degradation patterns
in the subfigures (b) and (e) reveal that the drop in maximum
PTE is minor, in most cases; usually much lower than 1%.
The rightmost graphs (c) and (f) show the deviation of the
optimal receiver capacitances (see Fig. 5(c)), obtained from
the original closed-form expressions (11) and (13). It can be
seen that the optimum capacitances remain almost unchanged
in all cases, particularly at larger distances.
Lastly, Fig. 10 confirms that the relaxations were tight in all
cases, as the normalized tightness errors always remained very
small, usually1 around 10−12 . Gaps in the error plots refer to
cases where the closed-form expressions did not lead to any
negative transmit powers (no power drawn from the system
via any of the transmit ports) and which, therefore, did not
have to be optimized using the relaxation method.
VI. S UMMARY & C ONCLUSION
A closed-form optimization framework [6] for multitransmitter (MISO) wireless power transfer systems has been
reviewed to highlight its simplicity and underlying physical
meanings. This powerful and intuitive set of equations is useful
for analysis and design of MISO and SISO WPT systems and
generalizes previous results for SISO systems [9].
However, as discussed, the obtained absolute optimum
operating modes can result in power being drawn via some
transmitter ports, rather than fed into the system. Although
analytically correct, such operating conditions are difficult to
realize in practice as power cannot easily be “recycled”.
1 Larger errors are due to the inherent numerical difficulties of the
impedance matrices at larger distances, particularly slight asymmetry.

10

To ensure practical realizability, it is important to incorporate constraints on the transmit powers into the optimization procedure to avoid such problems from the beginning.
However, this is not a trivial task, as such constraints are
nonconvex, thereby rendering the entire optimization problem
nonconvex.
This paper presents a step-by-step derivation of a convex
semidefinite relaxation of the nonconvex problem, where by
loosening some constraints a convex formulation is obtained,
which can efficiently and reliably be solved numerically by
dedicated algorithms. Furthermore, a simple test is given,
which reveals whether the solved problem is equivalent to
the original nonconvex problem (i.e. whether the relaxation
is tight) and that resulting optimum is indeed the true global
optimum looked for. This is a clear advantage over global
optimization methods, such as genetic algorithms (GA), where
convergence to the true global optimum cannot be guaranteed
or tested.
Simulation results of some basic multi-transmitter WPT
systems show that the new method is very practical and
indeed able to avoid any realizability issues due to negative
transmit powers. Furthermore, it is shown that the resulting
performance gap between the closed-form (absolute) optimum
and the new, practically realizable solution is usually very
small; typically below 1% of the predicted PTE. Lastly, plots
of the tightness test results reveal that in all cases under
consideration the relaxation is indeed tight, i.e. that the solved
problem is equivalent to the actual nonconvex problem and
that the obtained optimum PTE is the global optimum PTE
satisfying all constraints.
This new optimization procedure is powerful, versatile and
retains the rigor of convex optimization, while ensuring the
practical realizability of the optima it produces.

A PPENDIX A
A NALYTICAL F ORMULATION OF THE P OSITIVE AND
N EGATIVE PIM PARTS BY E IGENVECTORS
Let the impedance matrix of the unloaded WPT system be
Z = R + jωL + jωM ,

(44)

where R and L are real-valued diagonal matrices containing
the losses and self-reactances (usually inductances Ln > 0, for
loop-based magnetically coupled systems) of the transmitters
and the receiver. M = MT is a symmetric (due to reciprocity
of the passive system) hollow matrix containing the mutual
impedances jωMn,m . Generally, each Mn,m is complexvalued, due to retardation effects when the electrical distance
between the transmitter(s) and receiver are not very small.
Moreover, let
{λn,m , vn,m } = eig(Tn )

n = 1, ... , N

(45)

denote the mth eigenvalue λn,m and corresponding eigenvector vn,m of the N × N port impedance matrix (PIM) Tn . In

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES

z

11

3=0

3=0

30

30

!30

!30

θ

3=0

0.056
0.106
0.156

30

0.056
0.106
0.156

!30

y

60

x

60

!60

90
Pt = !20 !10

0

10

!90 90
20 W

"2max = !50

(a)
z

60

!60

!40

!30

!90 90
!20 dB

!60

106 #"Cr /Cr = !6

(b)

3=0
30

!30

0

!90
6

3

(c)

3=0

30

!3

!30

θ

3=0

0.056
0.106
0.156

30

!30

0.056
0.106
0.156

y

60

x

60

!60

90
Pt = !20 !10

0

10

60

!60

!90 90
20 W

"2max = !50

(d)

!40

!30

!90 90
!20 dB

!60

106 #"Cr /Cr = !40 !20

(e)

0

20

!90
40

(f)

Fig. 11. Transmit powers Pt,n of closed-form optima (thin, dashed) and SDR results (thick, solid) at distance d = 0.1λ (a, d), PTE reduction to closed-form
optimization ∆ηmax in dB (b, e), and normalized deviation of receiver capacitance ∆Cr /Cr (c, f) of the coaxial multi-transmitter configurations MISO-2c
(a-c) and MISO-3c (d-f).

z

3=0

z

3=0

30

30

!30

θ

!30

θ

y

y

60

x

90

0 = 10!16

10!12

30
z

!30
3=0

30

90
90

0 = 10!16

H
±
±
± H ±
(v±
n ) Tn vn = ±λn · (vn ) vn

30
!60

!30

(c)

θ
y

60

!90 90 !90
= !50
!30 !20 dB
0"2
= max
10!16
10!12!40 10!8

vH
n,m Tn vn,m

z

3=0

!60

60

!90
10!8

10!12

(b)

0.056
0.106
0.156

!30

60
y

x

x !60

!90 90
10!8

(a) 3 = 0

θ

60

!60

x !60

0 = 10

!16

index m, as it is clear from (46) that m = 1, 2 correspond to the superscripts −, +, respectively) and a superscript
zero point to eigenvectors corresponding to zero eigenvalues,
i.e. vn,m>2 = v0n,m (the index m starts at 3, for these
eigenvectors). Hence, the quadratic forms with respect to the
eigenvalues and eigenvectors are given by

10

!12

!90
10!8

(d)

Fig. 10. Normalized tightness errors of the optimal SDR solutions of the four
test cases, corresponding to Figs. 9 and 11. Apart from a few exceptions, the
erros are usually in the range of 10−12 , confirming that the solution vectors
represent the full semidefinite solution well and the relaxation is tight.

the order “negative, positive, and zero”, the eigenvalues can
be given as


p
1

2 + R2 < 0 m = 1
−λ−
R
−
S
=
n

n
n
n

2




p
1
λn,m =
λ+
Rn + Sn2 + Rn2 > 0 m = 2
n =


2



0
m = 3, ... , N
(46)
P
2
where the shorthand Sn2 = ω 2 m |Mn,m | > 0 has been
−
used. Note that, with these definitions, λ+
n , λn > 0, for all n.
Finally, let the eigenvectors corresponding to the non-zero
eigenvalues be denoted by vn,m=1,2 = v∓
n (omitting the

=0

∀ m 6= 1, 2 .

(47a)
(47b)

The PIMs can be separated into their positive and negative
(semidefinite) parts

T+
n

−
Tn = T+
n − Tn ,

, T−
n

(48)

 0. Further, each part is simply obtained
where both
from its eigenvalues and eigenvectors
±
T±
n = λn

± H
v±
n (vn )
± H ±
(vn ) vn

(49)

with the denominator being the outer product and the numerator the inner product of the respective eigenvectors.
These eigenvectors v±
n , can be obtained analytically as well:
 ± 
M̃n,1
1  . 
±
vn = ±  .. 
(50)
M̃n,N
±
M̃n,N

±
where M̃n,m
are the entries of the matrix M̃ which is identical
to the mutual impedance matrix M, as given in (44), with the
exception of the diagonal:
±

M̃ = M ± 2Diag(λ± ) .

(51)

Obtaining the positive and negative parts of the PIMs analytically, directly from the impedance matrix entries, adds

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both computational efficiency as well as numerical precision
as compared to using a numerical eigenvalue decomposition.
R EFERENCES
[1] N. Tesla, “High frequency oscillators for electro-therapeutic and other
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