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Analysis and optimisation of three-coil wireless power transfer systems
Article in IET Power Electronics · January 2018
DOI: 10.1049/iet-pel.2016.0492
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Analysis and Optimization of 3-Coil Wireless Power Transfer Systems
Paulo J. Abatti, Caio M. de Miranda*, Márcio A. P. da Silva and Sérgio F. Pichorim
Graduate School of Electrical Engineering and Computer Science (CPGEI) at the Federal University of
Technology – Paraná (UTFPR), Av. Sete de Setembro 3165, 80230-901 Curitiba, Brazil
*
caio.demiranda@gmail.com
Abstract: In this work the analysis of a 3-coil wireless power transfer (WPT) system, which can be divided in source,
communication and load circuits, is discussed in details. Among the 3-coil WPT systems features, it is demonstrated, for
instance, that maximum efficiency (𝜼𝑴𝑨𝑿 ) and maximum power transferred to the load (P3MAX) do not depend on the
load resistance, neither on the mutual inductance between communication and load coils. In fact, it is shown that 𝜼𝑴𝑨𝑿
and P3MAX depend only on source and communication circuits parameters. Practical results are also presented, showing
good agreement with the developed theory and validating the proposed analysis.
1. Introduction
There are several instances of electronic circuits for
which it is not recommended to connect the load and the
source directly using wires. Implantable biomedical devices
are certainly one of them, since transcutaneous wiring is
usually related to possible biological infection, among other
disadvantages [1]. One alternative to avoid wiring is to
supply these devices with the required power using the socalled wireless power transfer (WPT) system based on
inductive links. This choice is particularly adequate to
biomedical applications because of the relative high
conductivity of the biological tissue [2]. Therefore, it is not
a surprise to find works on design and optimization of
inductive links for biomedical applications, usually using a
2-coil configuration, as early as decade of 1970 [3].
More recently, general purpose 3 and 4-coil WPT
systems had been presented in the literature [4-12].
Following these pioneering works, these WPT systems had
been discussed in details, so that, most of their features had
been unraveled. For example, in [4-7,12] emphasis is given
to 4-coil systems showing that, compared with 2-coil WPT
systems, they have comparatively an augmented design
degree of freedom [8], so the manner to optimize the WPT
circuit performance increases proportionally.
Three-coil systems have been proposed in literature
[13-15]. For instance, in [14, 15] among other parameters,
the frequency-splitting phenomenon and change in resonant
frequency due to non-adjacent coupling between source
(transmitter) and load circuits are analyzed for 3-coil
systems. In addition, in [13] it is shown that 3-coil systems
can achieve a higher power deliver to the load compared
with 4-coil systems. In this way, this kind of circuit is
suitable for biomedical applications, in which the power
transfer is critical.
Implantable biomedical devices are usually designed
following strategies which include to implement the
implantable circuit as small as possible [16-21]. Of course,
the customized WPT system should in principle follow the
above rules. Additionally, some biomedical applications can
also involve larger power transfer, as is the case for artificial
organs [22, 23].
There are several manners one can use the WPT
system properties to further optimize an implantable
biomedical devices [24-26]. For example, in principle the
power consumption at the implantable unit can be predicted
by monitoring the source current in a WPT system [27].
This may be important considering that usually the
implantable device has limitations on size, and the
aforementioned property does not require in loco additional
circuitry close to the load.
Figure 1 shows 3-coil WPT systems customized to
implantable biomedical applications.
Fig. 1. Possible use of 3-coil WPT systems in implantable
biomedical devices, where L1 represents the source circuit
coil, L2 the communication circuit coil, and L3 the load
circuit coil.
It is important to emphasize that there are examples in the
literature which the communication circuit is placed together
with the load circuit (summarized by L2 and L3, respectively,
in Fig. 1) on the implantable part of the biomedical device
[13, 28]. However, according with the previous discussion,
it may apparently be better suited to implantable biomedical
applications to consider the load circuit the core of the
implantable unit and source and communication circuits
outside the body (see Fig. 1).
The aim of this paper is to present a short analysis
of 3-coil WPT systems, giving emphasis on system
properties, particularly those that may be of interest to
implantable biomedical circuit designers. Particularly, the
This paper is a postprint of a paper submitted to and accepted for publication
in IET Power Electronics and is subject to Institution of Engineering and
Technology Copyright. The copy of record is available at the IET Digital Library
main novelty here presented when compared with the
specialized literature, is that it is shown that when the nonadjacent coupling between source and load circuits can be
neglected, maximum efficiency ( 𝜂𝑀𝐴𝑋 ) and maximum
power transferred to the load (P3MAX) do not depend on the
load resistance, neither on the mutual inductance between
communication and load coils. In fact, it is shown that these
parameters depend only on source and communication
circuits parameters. Practical results are also presented
validating the presented theory.
2. Circuit Analysis
In general, the objective of any WPT system,
including the 3-coil one, is to transfer the maximum amount
of power from the source to the load, usually located as far
as possible, with maximum efficiency. However, it is well
known (maximum power transfer theorem) that maximum
power transfer is attained with an overall system efficiency
of only 50%, higher efficiencies means a relatively reduced
amount of power transferred to the load [5, 8]. Thus, it is
necessary to know a priori whether the WPT system is
primarily devised to supply the implantable device with
maximum possible amount of power or if the priority is to
optimize the system overall efficiency [5]. Moreover, in
most of the cases it is not possible to align the coils of the
implantable biomedical devices (coaxial arrangement) so
that it is not possible to relate WPT system performance
only with distance between coils (see Fig.1).
Figure 2 shows the schematic view of the 3-coil WPT
circuit. Independent on the WPT system application, it is
usual to consider all its circuits tuned at the same resonance
angular
frequency
(𝜔0 = (√𝐿1 𝐶1 )−1 = (√𝐿2 𝐶2 )−1 =
−1
respectively, where M12 and M23 are the remaining mutual
inductances, and v is the source open-terminal voltage
(when i1 = 0). In this way, P1 + P2 + P3 = v.i1 = PT (the total
power supplied by the voltage source).
The WPT system efficiency (𝜂) can be written as
𝜂=
𝑅3 𝜔0 2 𝑀12 2 𝜔0 2 𝑀23 2
2
2
(𝑅3 𝜔0 𝑀12 +𝑅1 (𝑅2 𝑅3 +𝜔0 2 𝑀23 2 ))(𝑅2 𝑅3 +𝜔0 2 𝑀23 2 )
.
(4)
Usually the designer objective is to optimize P3 and 𝜂
as a function of M23 and M12, respectively. Fortunately, due
to their format, (3) and (4) present a maximum as a function
of M23, whereas (3) presents a maximum as a function of
M12 (observe that (4) does not present a maximum as a
function of M12). Additionally, note that (3) and (4) present
also a maximum as a function of R3.
2.1 Influence of M12
The optimal value of M12 that maximizes the power
transferred to R3, can be obtained solving dP3/dM12 = 0,
yielding
𝑀12 =
1
√
𝜔0
(𝑅2 𝑅3 +𝜔0 2 𝑀23 2 )𝑅1
𝑅3
.
(5)
Substituting (5) in (1), (2), (3) and (4) gives
𝑃1 =
𝑃2 =
𝑃3 =
𝑣2
(6a)
,
4𝑅1
𝑣 2 𝑅2 𝑅3
,
(6b)
,
(6c)
4𝑅1 (𝑅2 𝑅3 +𝜔0 2 𝑀23 2 )
𝑣 2 𝜔0 2 𝑀23 2
4𝑅1 (𝑅2 𝑅3 +𝜔0 2 𝑀23 2 )
and
𝜂=
𝜔0 2 𝑀23 2
(6d)
,
2(𝑅2 𝑅3 +𝜔0 2 𝑀23 2 )
(√𝐿3 𝐶3 ) ) , and to neglect the influence of mutual
respectively.
inductance M13, because under these conditions the voltages
and currents in each circuit are in phase reducing possible
losses due to reactive effects. Moreover, for sake of
simplicity in the equations presentation, it is hereafter used
in the computations R3, which is the sum of internal
resistances r3 of the capacitance C3 and inductance L3 and
the load resistance RL, and R1, which is the sum of internal
resistances r1 of the capacitance C1 and inductance L1 and
the source internal resistance RS. Also, for coherence in the
equations presentation the losses in circuit 2 (r2 - the sum of
internal resistances of the capacitance C2 and inductance L2)
will be represented as R2. Finally, due to the aforementioned
implantable biomedical devices features, the WPT system it
is optimized as a function of the mutual inductances.
These considerations allow to write the power
dissipated at source (P1), communication (P2) and load (P3)
circuits as
Equations (6) indicate that P1 = P2 + P3 = 𝑣2 /4𝑅1 , so
that always 𝜂 ≤ ½, and the ideal maximum power transfer to
the load is PMAX = 𝑣2 /4𝑅1 , which would occur only if R2 =
0 (the ideal case with no losses in the communication
circuit).
2
2
𝑃1 = |𝑖1 | 𝑅1 =
𝑃2 = |𝑖2 |2 𝑅2 =
and
𝑃3 = |𝑖3 |2 𝑅3
𝑅1 (𝑅2 𝑅3 +𝜔0 2 𝑀23 2 )
2
(𝑅1 𝑅2 𝑅3 +𝑅1 𝜔0 2 𝑀23 2 +𝑅3 𝜔0 2 𝑀12 2 )
𝑅2 𝑅3 2 𝜔0 2 𝑀12 2
(𝑅1 𝑅2 𝑅3 +𝑅1 𝜔0 2 𝑀23 2 +𝑅3 𝜔0 2 𝑀12 2 )2
𝑅3 𝜔0 2 𝑀12 2 𝜔0 2 𝑀23 2
(𝑅1 𝑅2 𝑅3 +𝑅1 𝜔0 2 𝑀23 2 +𝑅3 𝜔0 2 𝑀12 2 )2
2
𝑣 ,
2.2 Influence of M23
The optimal value of M23 that maximizes the power
transferred to R3, can be obtained solving dP3/dM23 = 0,
yielding
𝑀23 =
𝑃1 =
(1)
𝑣2
(2)
(3)
𝜔0
√
𝑅1
.
(7)
Substituting (7) in (1), (2), (3) and (4) gives
𝑃2 =
𝑣2
(𝑅1 𝑅2 +𝜔0 2 𝑀12 2 )𝑅3
1
𝑃3 =
𝑣 2 (2𝑅1 𝑅2 +𝜔0 2 𝑀12 2 )2
4𝑅1 (𝑅1 𝑅2 +𝜔0 2 𝑀12 2 )2
𝑣 2 𝑅2 𝜔0 2 𝑀12 2
4(𝑅1 𝑅2 +𝜔0 2 𝑀12 2 )2
𝑣 2 𝜔0 2 𝑀12 2
,
(8a)
(8b)
,
,
4𝑅1 (𝑅1 𝑅2 +𝜔0 2 𝑀12 2 )
(8c)
and
𝜂=
𝜔0 2 𝑀12 2
2(2𝑅1 𝑅2 +𝜔0 2 𝑀12 2 )
,
(8d)
2
Fig. 2. Schematic representation of a 3-coil WPT system, where here also (1) represents the source circuit, (2) the
communication circuit, and (3) the load circuit.
interesting, from the circuit designer point of view, to note
respectively. Equations (8) indicate that always P1 ≥ P2 + P3,
that optimization of both power transfer and efficiency as a
function of M23 do not depend either on itself (M23) and R3.
so that always 𝜂 ≤ 1/2. Again, the ideal maximum power
2
This means that independent on R3 value the maximum
transfer to the load (PMAX = 𝑣 /4𝑅1 ) would occur only if R2
power transferred to the load or the system efficiency is
= 0.
exclusively determined by the source and communication
The optimal value of M23 that maximizes 𝜂, can be
circuits’ parameters. As this is suitable for the approach
obtained solving dη/dM23 = 0, yielding
shown in Fig. 1, where the source and communication
√(𝑅1 𝑅2 +𝜔0 2 𝑀12 2 )√𝑅1 𝑅2 𝑅3
1
circuits are outside the body, this design was experimentally
𝑀23 = √
.
(9)
𝜔0
𝑅1
tested.
Substituting (9) in (1), (2), (3) and (4) gives
It is important to clarify that in the previous
𝑅2 𝑣 2
equations
(4, 6, 8 and 10) if we look the efficiency
𝑃1 =
,
(10a)
𝑅1 𝑅2 +𝜔0 2 𝑀12 2
considering only the power delivered to the load (ηL), since
𝑣 2 𝑅2
𝜔0 2 𝑀12 2
the same current i3 flows through r3 and RL, the power P3
𝑃2 =
(10b)
2,
𝑅1 𝑅2 +𝜔0 2 𝑀12 2 (√(𝑅 𝑅 +𝜔 2 𝑀 2 )+ 𝑅 𝑅 )
can be splitted using the ratio of a voltage divider. Therefore
√ 1 2
1 2
0
12
PRL = P3 RL/(r3+RL) and the efficiency can be given by,
𝑣 2 √ 𝑅2
𝜔0 2 𝑀12 2
𝑅𝐿
𝑃3 =
,
(10c)
2
𝜂 =
. 𝜂.
(11)
2
2
√𝑅1 √𝑅1 𝑅2 +𝜔0 𝑀12 (√(𝑅1 𝑅2 +𝜔0 2 𝑀12 2 )+√𝑅1 𝑅2 )
𝐿
and
𝜔0 2 𝑀12 2
𝜂=
2
,
(10d)
(√(𝑅1 𝑅2 +𝜔0 2 𝑀12 2 )+√𝑅1 𝑅2 )
respectively.
The maximization of 𝜂 means, as already mentioned before,
that one is not maximizing 𝑃3 . This can be easily
demonstrated observing that 𝑃3 (see (10c)) presents a
maximum
at
𝜔0 2 𝑀12 2 = 2(√2 + 1)𝑅1 𝑅2
which
substituting in (10a), (10c) and (10d) gives 𝑃1 = 𝑃3 =
2
𝑣2 /(√2 + 1) 𝑅1 and 𝜂 = √2 − 1, i.e., 𝜂 ˂ 1/2.
Note in addition that if R2 = 0 (the already called
ideal case) the value of M23 which maximizes 𝜂 is zero (M23
= 0), with the obvious correspondent values for power of
zero (𝑃1 = 𝑃2 = 𝑃3 = 0). This can be easily understood
observing that, considering R2 = 0, (4) increases
monotonically as M23 is reduced (𝜂 → 𝜂𝑀𝐴𝑋 as M23→0), and
that (1), (2), and (3) decrease as M23 decrease (𝑃1 , 𝑃2 , 𝑃3 →
0 as M23→0).
2.3 Influence of R3
It is also easy to show that differentiating (4) and (3)
with respect to R3 give results equal to (7) and (9), of course
with the same consequences summarized by (8) and (10),
respectively. Observe that the WPT system optimization of
power transfer as a function of 𝑀12 shows results (see (6))
that do not depend on itself. However, it may be more
𝑅𝐿 +𝑟3
If only the link efficiency (ηLink, efficiency in the
transmission) is analyzed (excluding the generator resistance
Rs) it can be written as,
𝜂 = 𝜂𝐿𝑖𝑛𝑘
𝑅1 ′
𝑅1 ′ +𝑅𝑆
,
(12)
where R1' is the total impedance seen by the voltage source,
i.e., the sum of R1 and the reflected resistance [3] from
circuits 2 and 3 into circuit 1.
3. Experimental validation
For the experimental validation, 3 coils with 15 cm of
diameter, 22 turns of enameled copper wire of 19 AWG
were built. The self-inductances and their correspondent
series resistances (rs) of the coils are approximately 127 µH
and 3.3 Ω, in the frequency of 589 kHz, measured with
Agilent 4294A vector impedance analyzer. The circuits
were tuned by using commercial capacitors with nominal
value of 565 pF (the real values measured on the impedance
analyzer were considered) with a variable capacitor
(trimmer) in parallel, achieving the series resonance value of
589 kHz (the measured resistance of the capacitors in this
frequency is in the order of 10-3 Ω, and were neglected),
thus r1 = r2 = r3 = rs = 3.3 Ω. The mutual inductance in
function of the distances between two coaxially aligned
coils was numerically calculated using the classical
Neumann equation [16, 21, 29]. Therefore, measuring the
distance between adjacent coils gives the corresponding
mutual inductance value, facilitating the experiments (see
Fig.3).
3
In order to measure the relative power transfer and
efficiency, it is necessary to measure the currents in circuit 1
and circuit 3. Thus, a resistor (R0) of 10 Ω (the measured
resistance value was 9.85 Ω at 589 kHz) is used in series
with coil 1 in order to measure the current, and three
different loads (RL) of 10, 50, and 100 Ω (exact values of
9.82, 49 and 98.5 Ω) in circuit 3. All resistors presented
negligible stray inductances of 40 nH at 589 kHz. A
sinusoidal voltage signal (v) of 8.2 VRMS, 589 kHz was
applied on coil 1 (as mentioned before, this is the openterminal voltage) using a signal generator (Tektronix
AFG3101). To confirm that the resonant frequency has not
been changed by external influences of the setup, such as
cable capacitances and others, the resonant frequency was
confirmed by measuring a minimum voltage point over the
RLC (resistance, inductance and capacitance in primary)
source (primary) circuit (a single isolated coil) [30] using a
oscilloscope (Agilent MSO6030). The internal impedance of
the signal generator (RS = 50 Ω) must be taken into account,
since it is in series with coil 1 and contributes to the total
series resistance. Thus, it is possible to calculate the total
resistance R1 of 63.15 Ω (R0 + RS + r1). The total load
resistance considered is R3 = RL + r3.
Fig. 3. Theoretical mutual inductance in function of the
distance for the constructed coils with 15 cm of diameter,
and 22 turns of enameled copper wire.
The currents in primary circuit and load, i1 and i3,
respectively, could be found, as mentioned before, by
measuring the voltages over R0 and RL. Finally, the power
delivered to the system can be computed as PT = v.i1 and the
power in R3 using (3) as 𝑃3 = 𝑅3 ⋅ |𝑖3 |2 . Then the overall
efficiency (η) can be found as the ratio P3/PT, which is the
same as using (4). Of course, if only the power in the load
should be computed then equation (11) can be used. In a
similar way, the link efficiency can be computed by using
(12), where in this case, the series resistor R0 (used to
measure the current) can also be included as part of RS.
Following the strategy adopted in the last session, the
mutual inductance between coil 1 and 2 was kept constant at
M12 = 2.667 µH (distance of 18 cm) and M12 = 1.384 µH
(distance of 24 cm), while the mutual inductance between
coils 2 and 3 was gradually varied from 37.97 µH (distance
of 2 cm between coils 2 and 3) to 1.38 µH (distance of 24
cm between coils 2 and 3). In fact, the distance considered is
between the last turn of coil 2 and first turn coil 3. For the
power transfer evaluation it is interesting to compute a
relative power transfer [5], defined as the ratio P3/PMAX,
where, as mentioned PMAX = v2/4R1 (the maximum power
transfer when there are no losses in circuit 2). Therefore, the
relative power transfer and efficiency can be measured in
the considered mutual inductance range. The experimental
setup is shown in Fig. 4.
Fig. 4. Experimental setup consisting of 3 coils with 15 cm
of diameter, 22 turns of enameled copper wire of 19 AWG,
digital oscilloscope (shown in right side), and signal
generator.
The measured and calculated, relative power transfer
and efficiency, for the three load values are shown in Fig. 5
and 6. The smallest correlation coefficient for the three
loads, considering both efficiency and relative power
transfer is 0.998. Notice that using mutual inductance in
Fig.5 and 6 is a more generic presentation of the results than
to use distance between coils, since it can include different
coils design and coil misalignments. As mentioned before,
(7) and (9) show that maximum efficiency and maximum
power transfer occurs for different M23. For example, in Fig.
5 R3 = 101.8 Ω and M12 = 2.67 µH, then according to (7) and
(9) P3 is maximum for M23 = 6 µH and η is maximum for
M23 = 5.45 µH. Of course, this point depends on the value of
the product ωoM12 (notice that when it is zero (7) and (9) are
identical).
4. Conclusion
The results of Fig. 5 and 6 show that, as predicted in
theory, by correctly adjusting R3 and M23, both maximum
efficiency (𝜂𝑀𝐴𝑋 ) and maximum power transferred to the
load (P3MAX) depend only on M12. In fact, 𝜂𝑀𝐴𝑋 and P3MAX
increase as M12 increases. This features can be advantageous
in practical applications, for example, in applications
involving battery charging, it simplifies the necessary
electronic circuits, since any equivalent load resistance
variation can be compensated by adjusting M23 (it as in can
be seen in Fig. 5 and 6) so that efficiency or power transfer
can be kept at a same maximum value. It should be
emphasized that these features are only valid if all coils are
tuned at the same frequency, and M13 is negligible. It should
be emphasized that, to the authors best known, this features
were not presented in literature.
It is also important to highlight that, though a lowpower experimental setup was here used, the same
advantages and characteristics showed in this work can be
achieved for higher power applications.
4
i.e. independent on the load value (R3) sources with low
internal impedances should be used.
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loads.
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