See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/323112313 Analysis and optimisation of three-coil wireless power transfer systems Article in IET Power Electronics · January 2018 DOI: 10.1049/iet-pel.2016.0492 CITATIONS READS 17 597 4 authors, including: Paulo José Abatti Caio Marcelo de Miranda Federal University of Technology - Paraná/Brazil (UTFPR) Federal University of Technology - Paraná/Brazil (UTFPR) 60 PUBLICATIONS 418 CITATIONS 16 PUBLICATIONS 136 CITATIONS SEE PROFILE Sérgio Francisco Pichorim Federal University of Technology - Paraná/Brazil (UTFPR) 73 PUBLICATIONS 460 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Biomedical Instrument View project Biotelemetry View project All content following this page was uploaded by Caio Marcelo de Miranda on 26 April 2018. The user has requested enhancement of the downloaded file. SEE PROFILE 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 (πΌπ΄π¨πΏ ) 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 πΌπ΄π¨πΏ 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 ( πππ΄π ) 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 (π0 = (√πΏ1 πΆ1 )−1 = (√πΏ2 πΆ2 )−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 (π) can be written as π= π 3 π0 2 π12 2 π0 2 π23 2 2 2 (π 3 π0 π12 +π 1 (π 2 π 3 +π0 2 π23 2 ))(π 2 π 3 +π0 2 π23 2 ) . (4) Usually the designer objective is to optimize P3 and π 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 π12 = 1 √ π0 (π 2 π 3 +π0 2 π23 2 )π 1 π 3 . (5) Substituting (5) in (1), (2), (3) and (4) gives π1 = π2 = π3 = π£2 (6a) , 4π 1 π£ 2 π 2 π 3 , (6b) , (6c) 4π 1 (π 2 π 3 +π0 2 π23 2 ) π£ 2 π0 2 π23 2 4π 1 (π 2 π 3 +π0 2 π23 2 ) and π= π0 2 π23 2 (6d) , 2(π 2 π 3 +π0 2 π23 2 ) (√πΏ3 πΆ3 ) ) , 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 = π£2 /4π 1 , so that always π ≤ ½, and the ideal maximum power transfer to the load is PMAX = π£2 /4π 1 , which would occur only if R2 = 0 (the ideal case with no losses in the communication circuit). 2 2 π1 = |π1 | π 1 = π2 = |π2 |2 π 2 = and π3 = |π3 |2 π 3 π 1 (π 2 π 3 +π0 2 π23 2 ) 2 (π 1 π 2 π 3 +π 1 π0 2 π23 2 +π 3 π0 2 π12 2 ) π 2 π 3 2 π0 2 π12 2 (π 1 π 2 π 3 +π 1 π0 2 π23 2 +π 3 π0 2 π12 2 )2 π 3 π0 2 π12 2 π0 2 π23 2 (π 1 π 2 π 3 +π 1 π0 2 π23 2 +π 3 π0 2 π12 2 )2 2 π£ , 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 π23 = π1 = (1) π£2 (2) (3) π0 √ π 1 . (7) Substituting (7) in (1), (2), (3) and (4) gives π2 = π£2 (π 1 π 2 +π0 2 π12 2 )π 3 1 π3 = π£ 2 (2π 1 π 2 +π0 2 π12 2 )2 4π 1 (π 1 π 2 +π0 2 π12 2 )2 π£ 2 π 2 π0 2 π12 2 4(π 1 π 2 +π0 2 π12 2 )2 π£ 2 π0 2 π12 2 , (8a) (8b) , , 4π 1 (π 1 π 2 +π0 2 π12 2 ) (8c) and π= π0 2 π12 2 2(2π 1 π 2 +π0 2 π12 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 π ≤ 1/2. Again, the ideal maximum power 2 This means that independent on R3 value the maximum transfer to the load (PMAX = π£ /4π 1 ) 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 π, 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 √(π 1 π 2 +π0 2 π12 2 )√π 1 π 2 π 3 1 circuits are outside the body, this design was experimentally π23 = √ . (9) π0 π 1 tested. Substituting (9) in (1), (2), (3) and (4) gives It is important to clarify that in the previous π 2 π£ 2 equations (4, 6, 8 and 10) if we look the efficiency π1 = , (10a) π 1 π 2 +π0 2 π12 2 considering only the power delivered to the load (ηL), since π£ 2 π 2 π0 2 π12 2 the same current i3 flows through r3 and RL, the power P3 π2 = (10b) 2, π 1 π 2 +π0 2 π12 2 (√(π π +π 2 π 2 )+ π π ) 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, π£ 2 √ π 2 π0 2 π12 2 π πΏ π3 = , (10c) 2 π = . π. (11) 2 2 √π 1 √π 1 π 2 +π0 π12 (√(π 1 π 2 +π0 2 π12 2 )+√π 1 π 2 ) πΏ and π0 2 π12 2 π= 2 , (10d) (√(π 1 π 2 +π0 2 π12 2 )+√π 1 π 2 ) respectively. The maximization of π means, as already mentioned before, that one is not maximizing π3 . This can be easily demonstrated observing that π3 (see (10c)) presents a maximum at π0 2 π12 2 = 2(√2 + 1)π 1 π 2 which substituting in (10a), (10c) and (10d) gives π1 = π3 = 2 π£2 /(√2 + 1) π 1 and π = √2 − 1, i.e., π Λ 1/2. Note in addition that if R2 = 0 (the already called ideal case) the value of M23 which maximizes π is zero (M23 = 0), with the obvious correspondent values for power of zero (π1 = π2 = π3 = 0). This can be easily understood observing that, considering R2 = 0, (4) increases monotonically as M23 is reduced (π → πππ΄π as M23→0), and that (1), (2), and (3) decrease as M23 decrease (π1 , π2 , π3 → 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 π12 shows results (see (6)) that do not depend on itself. However, it may be more π πΏ +π3 If only the link efficiency (ηLink, efficiency in the transmission) is analyzed (excluding the generator resistance Rs) it can be written as, π = ππΏπππ π 1 ′ π 1 ′ +π π , (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 β¦, 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 β¦, and were neglected), thus r1 = r2 = r3 = rs = 3.3 β¦. 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 β¦ (the measured resistance value was 9.85 β¦ 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 β¦ (exact values of 9.82, 49 and 98.5 β¦) 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 β¦) 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 β¦ (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 π3 = π 3 ⋅ |π3 |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 β¦ 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 (πππ΄π ) and maximum power transferred to the load (P3MAX) depend only on M12. In fact, πππ΄π 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. 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