Abstract
In this paper, it is demonstrated that the efficiency and ability to transfer power to the load in threecoil wireless power transfer (WPT) systems are always higher than in equivalent fourcoil ones. On the other hand, it is shown that there are features attainable in fourcoil WPT system that are not in threecoil ones. For instance, in a fourcoil WPT system, which can be divided into source, two communication, and load circuits, it is possible to devise a method for which the maximum power transferred to the load circuit or the maximum efficiency do not depend on the mutual inductance between the two communication coils, independently of the load resistance value. The necessary conditions to achieve the above feature together with the overall circuit analysis are discussed in details and practical results presented.
Index Terms
fourcoil; power transfer efficiency; threecoil; wireless power transfer systems
I. INTRODUCTION
Among the several forms of energy, whenever possible, the electrical one is preferable as produces less pollution comparatively, it is easier to handle, and mainly because it can be transmitted more efficiently. The usual method to transmit electrical energy from the source to the load is via cables or wires. However, from the very beginning of electrical energy distribution history, it was recognized that wireless methods to transmit it would be comparatively more convenient [^{1}[1] A. Marincic, “Nikola Tesla and the Wireless Transmission of Energy,” IEEE Transactions on Power Apparatus and Systems, vol. PAS101, no. 10, pp. 4064–4068, oct 1982. [Online]. Available: http://ieeexplore.ieee.org/document/4111223/
http://ieeexplore.ieee.org/document/4111...
].
Nevertheless, after the pioneering work of Tesla, which used an inductive link, composed of two coils tuned at the same resonance frequency to transmit electrical energy at a given distance [^{2}[2] N. Tesla, “Apparatus for Transmitting Electrical Energy,” p. 4, 1914. [Online]. Available: http://large.stanford.edu/courses/2014/ph240/ho1/docs/US1119732.pdf
http://large.stanford.edu/courses/2014/p...
], the investigation of the socalled wireless power transfer (WPT) systems was almost neglected for several years, but by some sparse works [^{3}[3] W. H. Ko, S. P. Liang, and C. D. F. Fung, “Design of radiofrequency powered coils for implant instruments,” Medical Biological Engineering Computing, vol. 15, no. 6, pp. 634–640, nov 1977. [Online]. Available: http://link.springer.com/10.1007/BF02457921
http://link.springer.com/10.1007/BF02457...
]–[^{9}[9] G. Wang, W. Liu, M. Sivaprakasam, and G. Kendir, “Design and analysis of an adaptive transcutaneous power telemetry for biomedical implants,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 10, pp. 2109–2117, oct 2005. [Online]. Available: http://ieeexplore.ieee.org/document/1519624/
http://ieeexplore.ieee.org/document/1519...
]. In fact, only about a decade ago the three [^{10}[10] M. Kiani, UeiMing Jow, and M. Ghovanloo, “Design and Optimization of a 3Coil Inductive Link for Efficient Wireless Power Transmission,” IEEE Transactions on Biomedical Circuits and Systems, vol. 5, no. 6, pp. 579–591, dec 2011. [Online]. Available: http://ieeexplore.ieee.org/document/5951804/
http://ieeexplore.ieee.org/document/5951...
]–[^{18}[18] R. Lu, M. R. Haider, and Y. Massoud, “A ThreeCoil Coupled HighEfficiency Power Link for Wireless Power Transfer Application,” in 2019 IEEE 20th Wireless and Microwave Technology Conference (WAMICON), pp. 1–4, apr 2019. [Online]. Available: https://ieeexplore.ieee.org/document/8765470/
https://ieeexplore.ieee.org/document/876...
], and fourcoil [^{19}[19] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, “Wireless Power Transfer via Strongly Coupled Magnetic Resonances,” Science, vol. 317, no. 5834, pp. 83–86, jul 2007. [Online]. Available: http://www.sciencemag.org/cgi/doi/10.1126/science.1143254
http://www.sciencemag.org/cgi/doi/10.112...
]–[^{30}[30] C. M. Miranda, S. F. Pichorim, and P. J. Abatti, “On the impact of relay circuit losses in fourcoil wireless power transfer systems,” International Journal of Circuit Theory and Applications, vol. 47, no. 12, pp. 1922–1932, dec 2019. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/cta.2685
https://onlinelibrary.wiley.com/doi/abs/...
] WPT systems had been introduced. WPT systems using more than four coils had also been investigated, but most of the research effort in the area had been focused in the three and fourcoil configurations [^{25}[25] S. Y. R. Hui, W. Zhong, and C. K. Lee, “A Critical Review of Recent Progress in MidRange Wireless Power Transfer,” IEEE Transactions on Power Electronics, vol. 29, no. 9, pp. 4500–4511, sep 2014. [Online]. Available: http://ieeexplore.ieee.org/document/6472081/
http://ieeexplore.ieee.org/document/6472...
], [^{31}[31] A. K. RamRakhyani and G. Lazzi, “Multicoil approach to reduce electromagnetic energy absorption for wirelessly powered implants,” Healthcare Technology Letters, vol. 1, no. 1, pp. 21–25, jan 2014. [Online]. Available: https://digitallibrary.theiet.org/content/journals/10.1049/htl.2013.0035
https://digitallibrary.theiet.org/conte...
], [^{32}[32] J. Liu, X. Zhang, J. Yu, Z. Xu, and Z. Ju, “Performance Analysis for the Magnetically Coupled Resonant Wireless Energy Transmission System,” Complexity, vol. 2019, pp. 1–13, nov 2019. [Online]. Available: https://www.hindawi.com/journals/complexity/2019/6090427/
https://www.hindawi.com/journals/complex...
]. Here it is important to emphasize that the three and fourcoil WPT systems are, in some aspects, similar to the twocoil WPT systems, e.g., they have one coil connected to the source and one connected to the load. The differences are that the threecoil WPT systems have one additional (communication) coil and the fourcoil WPT systems have two additional (communication) coils. Moreover, following Tesla's original approach [^{2}[2] N. Tesla, “Apparatus for Transmitting Electrical Energy,” p. 4, 1914. [Online]. Available: http://large.stanford.edu/courses/2014/ph240/ho1/docs/US1119732.pdf
http://large.stanford.edu/courses/2014/p...
], all coils are tuned at the same resonance frequency and mutual inductance of nonadjacent coils are made as small as possible.
Anyway, perhaps because it is a relatively recent circuit configuration, the three and fourcoil WPT systems characteristics are still object of studies. For example, in a recent paper it was demonstrated that in a threecoil WPT system both the maximum efficiency (η3_{max}) and maximum power transferred to the load (P_{3}_{max}) depend on neither the mutual inductance between the coils of the communication and load circuits nor the load resistance value (R_{L}) [^{17}[17] P. J. Abatti, C. M. de Miranda, M. A. da Silva, and S. F. Pichorim, “Analysis and optimisation of threecoil wireless power transfer systems,” IET Power Electronics, vol. 11, no. 1, pp. 68–72, jan 2018. [Online]. Available: https://digitallibrary.theiet.org/content/journals/10.1049/ietpel.2016.0492
https://digitallibrary.theiet.org/conte...
]. This means thatη_{3}_{max} and P_{3}_{max} are only determined by the source and communication circuits parameters, a feature that may be relevant to those involved in the circuit implementation. However, this also means that given a load resistance value there is only one value of the mutual inductance between the coils of the communication and load circuits, and viceversa, for which either the maximum power transferred to the load circuit or the efficiency are maximum, restricting its practical application.
The aim of this work is to show that in fourcoil WPT systems the maximum efficiency or maximum power transferred to the load do not depend on mutual inductance between the coils of the communication circuits (M23) independently of the load resistance value, and viceversa. This is done by adjusting the mutual inductance between the coil at the last communication circuit and that at the load circuit (M34). In order to demonstrate this feature it is important to compare the three and fourcoil WPT systems, for it is demonstrated that the efficiency and the ability to transfer power to the load in threecoil WPT systems are always higher than in equivalent fourcoil ones. Thus, the mutual inductance (proportional to distance in a coaxial arrangement) between the coils of the communication circuits were preserved in both three and fourcoil WPT systems. The necessary conditions to attain the above feature as well as the overall circuit analysis are discussed in details and experimental results, used to validate the theoretical analysis, presented.
II. CIRCUIT ANALYSIS
Figure 1 shows the schematic view of a fourcoil WPT system. Following Tesla's original approach [^{2}[2] N. Tesla, “Apparatus for Transmitting Electrical Energy,” p. 4, 1914. [Online]. Available: http://large.stanford.edu/courses/2014/ph240/ho1/docs/US1119732.pdf
http://large.stanford.edu/courses/2014/p...
], all circuits should be tuned at the same resonance angular frequency (
These considerations allow to write the power dissipated at the load circuit (P_{4}) as
where M_{12}, M_{23}, and M_{34} are the remaining mutual inductances, ν the source openterminal voltage (when i_{1} = 0), R_{1} the sum of the source resistance and the total internal resistance of L_{1} and C_{1}(R_{1} = R_{s} + r_{1}), r_{2} and r_{3} the total internal resistances of L_{2} and C_{2}, and L_{3} and C_{3}, respectively, and R_{4} the sum of the load resistance and the total internal resistance of L_{4} and C_{4} (R_{4} = R_{L} + r_{4}).
The total power supplied by the voltage source can be easily calculated (P_{T} = ν.i_{1}) giving
Thus, the system efficiency (η = P_{4}/P_{T}) can be written as
It is important to emphasize that if one calculates the efficiency considering only the power delivered to the load (η_{L}), since the same current i_{4} flows through r_{4} and R_{L}, the power P_{4} can be splitted using the ratio of a voltage divider. Thus, P_{RL} = P_{4}.R_{L}/(R_{L} + r_{4}) and the efficiency is η_{L} = η.R_{L}/(R_{L} + r_{4}). In a similar manner, if only the efficiency of the link transmission (η_{LINK}) is to be analyzed (excluding the generator resistance, R_{S}), it can be written
http://ieeexplore.ieee.org/document/6292...
] from communication and load circuits into the source circuit. Moreover, at first glance, the WPT systems should be designed to transmit the maximum amount of power from the source to the load (located as far as possible) with maximum efficiency. However, the maximum power transfer theorem teaches that the maximum transference of power is attained with an overall system efficiency of only 50%, higher efficiencies meaning a relatively reduced amount of power transferred to the load [^{25}[25] S. Y. R. Hui, W. Zhong, and C. K. Lee, “A Critical Review of Recent Progress in MidRange Wireless Power Transfer,” IEEE Transactions on Power Electronics, vol. 29, no. 9, pp. 4500–4511, sep 2014. [Online]. Available: http://ieeexplore.ieee.org/document/6472081/
http://ieeexplore.ieee.org/document/6472...
], [^{26}[26] P. J. Abatti, S. F. Pichorim, and C. M. de Miranda, “Maximum Power Transfer versus Efficiency in MidRange Wireless Power Transfer Systems,” Journal of Microwaves, Optoelectronics and Electromagnetic Applications, vol. 14, no. 1, pp. 97–109, jun 2015. [Online]. Available: http://www.scielo.br/scielo.php?script=sci_arttext&pid=S217910742015000100097&lng=en&tlng=en
http://www.scielo.br/scielo.php?script=s...
], [^{34}[34] P. Silvester, Modern Electromagnetic Fields, N. Marcuvitz, Ed., Englewood Cliffs, NJ, 1968. [Online]. Available: https://archive.org/details/ModernElectromagneticFields/page/n173
https://archive.org/details/ModernElectr...
]. Thus, it is necessary to know a priori whether the WPT system is designed to optimize efficiency or if the amount of power transferred to load is to be the maximum [^{26}[26] P. J. Abatti, S. F. Pichorim, and C. M. de Miranda, “Maximum Power Transfer versus Efficiency in MidRange Wireless Power Transfer Systems,” Journal of Microwaves, Optoelectronics and Electromagnetic Applications, vol. 14, no. 1, pp. 97–109, jun 2015. [Online]. Available: http://www.scielo.br/scielo.php?script=sci_arttext&pid=S217910742015000100097&lng=en&tlng=en
http://www.scielo.br/scielo.php?script=s...
].
Anyway, in order to help a comparative analysis, figure 2 shows the schematic view of the threecoil WPT system. Observe that the fourcoil WPT system can be transformed into a threecoil equivalent one, reflecting R_{4} [^{33}[33] D. M. Beams and S. G. Annam, “Validation of a reflectedimpedance design method for wireless power transfer applications,” in 2012 IEEE 55th International Midwest Symposium on Circuits and Systems (MWSCAS), pp. 758–761, aug 2012. [Online]. Available: http://ieeexplore.ieee.org/document/6292131/
http://ieeexplore.ieee.org/document/6292...
] into the second communication circuit (see figure 2). In other words, both threecoil and fourcoil WPT systems are equivalent whenever (see figure 2)
In addition, it is possible to define
so that the power transferred to R_{3} in a threecoil WPT system (P_{3}) and efficiency (η_{3}) can be given by
and
respectively.
Dividing (1) by (6) and (3) by (7), and using (4) and (5) yield
Therefore, the threecoil WPT systems always present better performance than the fourcoil ones (P_{3} > P_{4} and η_{3} > η_{4}).
However, there are situations that performance should be relegated to a second plan to attend some practical demand. For instance, it can be easily demonstrated that in the threecoil WPT system the M_{23} for maximum power transferred to R_{3} (
https://digitallibrary.theiet.org/conte...
] as
and
respectively.
Substituting (9) and (10) into (6) and (7) yield
and
respectively.
Note that, as already pointed out in [^{17}[17] P. J. Abatti, C. M. de Miranda, M. A. da Silva, and S. F. Pichorim, “Analysis and optimisation of threecoil wireless power transfer systems,” IET Power Electronics, vol. 11, no. 1, pp. 68–72, jan 2018. [Online]. Available: https://digitallibrary.theiet.org/content/journals/10.1049/ietpel.2016.0492
https://digitallibrary.theiet.org/conte...
], (11) and (12) are independent on either M_{23} and R_{3}, i.e., P_{3}_{MAX} or η_{3}_{MAX} are determined exclusively by the source and communication circuits’ parameters. However, (9) and (10) show also that for a given R_{3}, and consequently for a given load
On the other hand, in fourcoil WPT systems, using (4) and (5) into (9), and substituting (11) into (8), and using R_{4} = r_{4} + R_{L}, yield
and
respectively, whereas using (4) and (5) into (10), and substituting (12) into (8), and also using R_{4} = r_{4} + R_{L}, give
and
respectively.
Observe that independently of R_{L} used, the value of M_{34} might be adjusted so that an adequate value of M_{23} may be obtained, allowing P_{4} or η_{4} to be maximum. In other words, in a fourcoil WPT system the M_{34} can be used as an "impedance match" circuit desvinculating the actual R_{L} value from the determination of M_{23} which allows P_{4} orη_{4} to be maximum.
III. EXPERIMENTAL RESULTS
For the experimental evaluation of the mathematical analysis, four coils with equal dimensions and shapes were built. The coils are circular with diameter of 150 mm and 22 mm of length, wound with 23 turns of enameled copper 20 AWG wire in a single layer way. The coils have selfinductance of 138.67 ± 0.21 μH with internal resistances of 3.41 ± 0.09Ω. All measurements were made using an Agilent precision vector impedance analyzer (model 4294A) operating at 552kHz. In order to obtain the practical value of the mutual inductance the coils were arranged coaxially, the value of the coupling coefficient (k) was measured, and then using
http://www.scielo.br/scielo.php?script=s...
]. Figure 3 shows the measured mutual inductance as a function of the distance between the coils coaxially aligned.
Commercial capacitors of 560pF were used to tune the circuits (the practical values was 556 ± 7 pF), with a variable capacitor (trimmer) in parallel, achieving the series resonance value of 552kHz. This frequency has been selected due to its handiness in tuning the circuits, and because it does not present adverse health effects [^{35}[35] International Commission on NonIonizing Radiation Protection (ICNIRP), “Guidelines for Limiting Exposure to Electromagnetic Fields (100 kHz to 300 GHz),” Health Physics, vol. 118, no. 5, pp. 483–524, may 2020. [Online]. Available: http://journals.lww.com/10.1097/HP.0000000000001210
http://journals.lww.com/10.1097/HP.00000...
], [^{36}[36] International Commission on NonIonizing Radiation Protection (ICNIRP), “Principles for NonIonizing Radiation Protection,” Health Physics, vol. 118, no. 5, pp. 477–482, may 2020. [Online]. Available: http://journals.lww.com/10.1097/HP.0000000000001252
http://journals.lww.com/10.1097/HP.00000...
]. The resistances of the capacitors at 552kHz were neglected because they were in order of milliohms.
The measured (at 552kHz) values of the load (R_{L}) used in the experiments were 5.67Ω, 8.24Ω, 9.97Ω, 12.03Ω, 17.98Ω, 21.77Ω and 46.95Ω. The parasitic selfinductance of the resistors were neglected because they were in order of nanohenry.
Figure 4 shows the implemented fourcoil WPT system. The coil of the source circuit was fixed to the left end of a wood support, whereas the second coil was fixed 12.5cm apart. The value of M_{12} was 6.η18 μH (see figure 3). A sinusoidal voltage signal (ν) of 7.1 V_{RMS} with a frequency of 552kHz, internal resistance (R_{s}) of 50.53Ω (Rigol signal generator  DG1022) was used as the voltage source. The current at the source circuit (i_{1}) was determined to measure the voltage at a series resistor (r = 1.02Ω). Therefore, the value of R_{1} (= r_{1} + r + R_{s}) used in the calculations was 54.89Ω. During the experiments the phase between ν and i_{1} was continuously monitored (ideally it must be zero) to certify that the influence of M_{13}, M_{14}, and M_{24} could in fact be neglected.
The value of i_{3} in the threecoil and i_{4} in the fourcoil WPT systems, respectively, were determined by measuring the voltages at the used loads, and the powers at the load circuits (P_{3} = R_{3}.i_{3}^{2} and P_{4} = R_{4}.i_{4}^{2}) were calculated.
From equations (14) and (16) it can be defined as a multiplying factor (F)
The maximum power transferred to the load circuit (P_{4}_{MAX}), and the maximum efficiency (η_{4}_{MAX}), both as a function of R_{L} for F equal to 1/2, 2/3 and 5/6, are shown in figures 5(a) and 5(b), respectively. Firstly, the experiments were performed keeping M_{23} fixed at 3.4μH. Then, just to check the independence between R_{L} and M_{23} the experiments were repeated keeping M_{23} fixed at 5.55μH. In addition, for comparison purposes, the values of P_{3}_{MAX} and η_{3}_{MAX} for M_{23} = 3.4μH and M_{23} = 5.55μH were also plotted in figures 5(a) and 5(b), respectively.
Experimental results of (a) maximum power transferred to the load circuit and (b) maximum efficiency, both as a function of R_{L} in a fourcoil WPT system. For comparison purposes the values of P_{3}_{MAX} and η_{3}_{MAX} for M_{23} = 3.4μH and M_{23} = 5.55μH were also plotted in figures 5(a) and 5(b), respectively.
Evidently, in the fourcoil WPT system each time R_{L} was changed the relative position of L_{4} was modified so that (
IV. CONCLUSION
The three and fourcoil WPT systems have been compared, showing that in the fourcoil ones neither the maximum power transferred to the load nor the maximum efficiency depends on the mutual inductance regardless of the on load resistance value, provided (
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Publication Dates

Publication in this collection
03 Mar 2021 
Date of issue
Mar 2021
History

Received
17 June 2020 
Reviewed
20 June 2020 
Accepted
19 Jan 2021