Comparison Between Three- and Four-coil Wireless Power Transfer Systems with Resonant Coils

Fábio B. de Moraes Paulo J. Abatti About the authors

Abstract

In this paper, it is demonstrated that the efficiency and ability to transfer power to the load in three-coil wireless power transfer (WPT) systems are always higher than in equivalent four-coil ones. On the other hand, it is shown that there are features attainable in four-coil WPT system that are not in three-coil ones. For instance, in a four-coil 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
four-coil; power transfer efficiency; three-coil; 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. PAS-101, 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 so-called 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 radio-frequency 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, Uei-Ming Jow, and M. Ghovanloo, “Design and Optimization of a 3-Coil 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 Three-Coil Coupled High-Efficiency 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 four-coil [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 four-coil 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 four-coil configurations [25[25] S. Y. R. Hui, W. Zhong, and C. K. Lee, “A Critical Review of Recent Progress in Mid-Range 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, “Multi-coil 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://digital-library.theiet.org/content/journals/10.1049/htl.2013.0035
https://digital-library.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 four-coil WPT systems are, in some aspects, similar to the two-coil WPT systems, e.g., they have one coil connected to the source and one connected to the load. The differences are that the three-coil WPT systems have one additional (communication) coil and the four-coil 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 non-adjacent coils are made as small as possible.

Anyway, perhaps because it is a relatively recent circuit configuration, the three- and four-coil WPT systems characteristics are still object of studies. For example, in a recent paper it was demonstrated that in a three-coil WPT system both the maximum efficiency (η3max) and maximum power transferred to the load (P3max) depend on neither the mutual inductance between the coils of the communication and load circuits nor the load resistance value (RL) [17[17] P. J. Abatti, C. M. de Miranda, M. A. da Silva, and S. F. Pichorim, “Analysis and optimisation of three-coil wireless power transfer systems,” IET Power Electronics, vol. 11, no. 1, pp. 68–72, jan 2018. [Online]. Available: https://digital-library.theiet.org/content/journals/10.1049/iet-pel.2016.0492
https://digital-library.theiet.org/conte...
]. This means thatη3max and P3max 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 vice-versa, 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 four-coil 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 vice-versa. 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 four-coil WPT systems, for it is demonstrated that the efficiency and the ability to transfer power to the load in three-coil WPT systems are always higher than in equivalent four-coil 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 four-coil 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 four-coil 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 (ω01=L1C1=L2C2=L3C3=L4C4), and the mutual inductances between non-adjacent coils should be as small as possible (M13 = M14 = M24 = 0). Under the above conditions, the currents and voltages at each coil circuit are in phase so that possible losses due to reactive effects are reduced.

Fig. 1
Schematic representation of a four-coil wireless power transfer system.

These considerations allow to write the power dissipated at the load circuit (P4) as

(1) P 4 = R 4 | i 4 | 2 = R 4 υ 2 ω 0 2 M 12 2 ω 0 2 M 23 2 ω 0 2 M 34 2 ( ( R 1 r 2 + ω 0 2 M 12 2 ) ( r 3 R 4 + ω 0 2 M 34 2 ) + R 1 R 4 ω 0 2 M 23 2 ) 2 ,

where M12, M23, and M34 are the remaining mutual inductances, ν the source open-terminal voltage (when i1 = 0), R1 the sum of the source resistance and the total internal resistance of L1 and C1(R1 = Rs + r1), r2 and r3 the total internal resistances of L2 and C2, and L3 and C3, respectively, and R4 the sum of the load resistance and the total internal resistance of L4 and C4 (R4 = RL + r4).

The total power supplied by the voltage source can be easily calculated (PT = ν.i1) giving

(2) P T = r 2 r 3 R 4 + r 2 ω 0 2 M 34 2 + R 4 ω 0 2 M 23 2 ( R 1 r 2 + ω 0 2 M 12 2 ) ( r 3 R 4 + ω 0 2 M 34 2 ) + R 1 R 4 ω 0 2 M 23 2 .

Thus, the system efficiency (η = P4/PT) can be written as

(3) η = R 4 ω 0 2 M 12 2 ω 0 2 M 23 2 ω 0 2 M 34 2 ( ( R 1 r 2 + ω 0 2 M 12 2 ) ( r 3 R 4 + ω 0 2 M 34 2 ) + R 1 R 4 ω 0 2 M 23 2 ) ( r 2 r 3 R 4 + r 2 ω 0 2 M 34 2 + R 4 ω 0 2 M 23 2 )

It is important to emphasize that if one calculates the efficiency considering only the power delivered to the load (ηL), since the same current i4 flows through r4 and RL, the power P4 can be splitted using the ratio of a voltage divider. Thus, PRL = P4.RL/(RL + r4) and the efficiency is ηL = η.RL/(RL + r4). In a similar manner, if only the efficiency of the link transmission (ηLINK) is to be analyzed (excluding the generator resistance, RS), it can be written η=ηLINK·R1*/(RS+R1*), where R1* is the sum of r1 and the reflected resistance [33[33] D. M. Beams and S. G. Annam, “Validation of a reflected-impedance 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...
] 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 Mid-Range 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=S2179-10742015000100097&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=S2179-10742015000100097&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 three-coil WPT system. Observe that the four-coil WPT system can be transformed into a three-coil equivalent one, reflecting R4 [33[33] D. M. Beams and S. G. Annam, “Validation of a reflected-impedance 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 three-coil and four-coil WPT systems are equivalent whenever (see figure 2)

Fig. 2
Schematic representation of a three-coil wireless power transfer system.
(4) R L * = ω 0 2 M 34 2 R 4 .

In addition, it is possible to define

(5) R 3 = r 3 + R L * ,

so that the power transferred to R3 in a three-coil WPT system (P3) and efficiency (η3) can be given by

(6) P 3 = R 3 | i 3 | 2 = R 3 υ 2 ω 0 2 M 12 2 ω 0 2 M 23 2 ( ( R 1 r 2 + ω 0 2 M 12 2 ) R 3 + R 1 ω 0 2 M 23 2 ) 2 ,

and

(7) η 3 = R 3 ω 0 2 M 12 2 ω 0 2 M 23 2 ( ( R 1 r 2 + ω 0 2 M 12 2 ) R 3 + R 1 ω 0 2 M 23 2 ) ( r 2 R 3 + ω 0 2 M 23 2 ) ,

respectively.

Dividing (1) by (6) and (3) by (7), and using (4) and (5) yield

(8) P 4 P 3 = η 4 η 3 = ω 0 2 M 34 2 R 4 r 3 + ω 0 2 M 34 2 R 4 .

Therefore, the three-coil WPT systems always present better performance than the four-coil ones (P3 > P4 and η3 > η4).

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 three-coil WPT system the M23 for maximum power transferred to R3 (M23P3MAX) and M23 for maximum efficiency (M− η3MAX) can be written [17[17] P. J. Abatti, C. M. de Miranda, M. A. da Silva, and S. F. Pichorim, “Analysis and optimisation of three-coil wireless power transfer systems,” IET Power Electronics, vol. 11, no. 1, pp. 68–72, jan 2018. [Online]. Available: https://digital-library.theiet.org/content/journals/10.1049/iet-pel.2016.0492
https://digital-library.theiet.org/conte...
] as

(9) M 23 P 3 M A X = 1 ω 0 R 1 r 2 + ω 0 2 M 12 2 R 1 R 3 ,

and

(10) M 23 η 3 M A X = 1 ω 0 r 2 R 1 R 1 r 2 + ω 0 2 M 12 2 R 3 ,

respectively.

Substituting (9) and (10) into (6) and (7) yield

(11) P 3 M A X = υ 2 4 R 1 ω 0 2 M 12 2 R 1 r 2 + ω 0 2 M 12 2 ,

and

(12) η 3 M A X = ω 0 2 M 12 2 ( R 1 r 2 + R 1 r 2 + ω 0 2 M 12 2 ) 2 ,

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 three-coil wireless power transfer systems,” IET Power Electronics, vol. 11, no. 1, pp. 68–72, jan 2018. [Online]. Available: https://digital-library.theiet.org/content/journals/10.1049/iet-pel.2016.0492
https://digital-library.theiet.org/conte...
], (11) and (12) are independent on either M23 and R3, i.e., P3MAX or η3MAX are determined exclusively by the source and communication circuits’ parameters. However, (9) and (10) show also that for a given R3, and consequently for a given load RL*=RL, there is only one value of M23 for which P3 or η3 can be maximum, and this specific value of M23 may not be attainable.

On the other hand, in four-coil WPT systems, using (4) and (5) into (9), and substituting (11) into (8), and using R4 = r4 + RL, yield

(13) M 23 P 4 M A X = 1 ω ( R 1 r 2 + ω 0 2 M 12 2 R 1 ) ( r 3 + ω 0 2 M 34 2 r 4 + R L ) ,

and

(14) P 4 M A X = υ 2 4 R 1 ω 0 2 M 12 2 R 1 r 2 + ω 0 2 M 12 2 ω 0 2 M 34 2 r 4 + R L r 3 + ω 0 2 M 34 2 r 4 + R L ,

respectively, whereas using (4) and (5) into (10), and substituting (12) into (8), and also using R4 = r4 + RL, give

(15) M 23 η 4 M A X = 1 ω 0 r 2 R 1 R 1 r 2 + ω 0 2 M 12 2 ( r 3 + ω 0 2 M 34 2 r 4 + R L ) ,

and

(16) η 4 M A X = ω 0 2 M 12 2 ( R 1 r 2 + R 1 r 2 + ω 0 2 M 12 2 ) 2 ω 0 2 M 34 2 r 4 + R L r 3 + ω 0 2 M 34 2 r 4 + R L ,

respectively.

Observe that independently of RL used, the value of M34 might be adjusted so that an adequate value of M23 may be obtained, allowing P4 or η4 to be maximum. In other words, in a four-coil WPT system the M34 can be used as an "impedance match" circuit desvinculating the actual RL value from the determination of M23 which allows P4 orη4 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 self-inductance 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 Mps=kLpLs the mutual inductance was determined as follows: the primary coil was excited by a signal generator (Rigol model DG1022) with a voltage vp, whereas the open-terminal voltage of the secondary coil, vs, was taken. Both voltages were measured with the aid of a digital oscilloscope (Tektronix model TDS2012C). The frequency of the exciting voltage was adjusted to a relatively low value (≈ 10kHz) to reduce the possible influence of the coils’ stray capacitances. It can be easily demonstrated that k = νp/νs, whenever LpLs [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=S2179-10742015000100097&lng=en&tlng=en
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.

Fig. 3
Experimental mutual inductance in function of distance for 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 Non-Ionizing 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 Non-Ionizing Radiation Protection (ICNIRP), “Principles for Non-Ionizing 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 (RL) used in the experiments were 5.67Ω, 8.24Ω, 9.97Ω, 12.03Ω, 17.98Ω, 21.77Ω and 46.95Ω. The parasitic self-inductance of the resistors were neglected because they were in order of nanohenry.

Figure 4 shows the implemented four-coil 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 M12 was 6.η18 μH (see figure 3). A sinusoidal voltage signal (ν) of 7.1 VRMS with a frequency of 552kHz, internal resistance (Rs) of 50.53Ω (Rigol signal generator - DG1022) was used as the voltage source. The current at the source circuit (i1) was determined to measure the voltage at a series resistor (r = 1.02Ω). Therefore, the value of R1 (= r1 + r + Rs) used in the calculations was 54.89Ω. During the experiments the phase between ν and i1 was continuously monitored (ideally it must be zero) to certify that the influence of M13, M14, and M24 could in fact be neglected.

Fig. 4
Experimental setup of the four-coil wireless power transfer system.

The value of i3 in the three-coil and i4 in the four-coil WPT systems, respectively, were determined by measuring the voltages at the used loads, and the powers at the load circuits (P3 = R3.|i3|2 and P4 = R4.|i4|2) were calculated.

From equations (14) and (16) it can be defined as a multiplying factor (F)

(17) F = ω 0 2 M 34 2 r 4 + R L r 3 + ω 0 2 M 34 2 r 4 + R L .

The maximum power transferred to the load circuit (P4MAX), and the maximum efficiency (η4MAX), both as a function of RL 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 M23 fixed at 3.4μH. Then, just to check the independence between RL and M23 the experiments were repeated keeping M23 fixed at 5.55μH. In addition, for comparison purposes, the values of P3MAX and η3MAX for M23 = 3.4μH and M23 = 5.55μH were also plotted in figures 5(a) and 5(b), respectively.

Fig. 5
Experimental results of (a) maximum power transferred to the load circuit and (b) maximum efficiency, both as a function of RL in a four-coil WPT system. For comparison purposes the values of P3MAX and η3MAX for M23 = 3.4μH and M23 = 5.55μH were also plotted in figures 5(a) and 5(b), respectively.

Evidently, in the four-coil WPT system each time RL was changed the relative position of L4 was modified so that ((ω02M342)/(RL+r4)) was kept constant.

IV. CONCLUSION

The three- and four-coil WPT systems have been compared, showing that in the four-coil 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 ((ω2M342)/(RL+r4)) is kept constant. Although the maximum power transferred to the load or maximum efficiency of four-coil are always smaller than those of three-coil WPT systems, the demonstrated feature allows designing optimized WPT systems independent on load resistance value whenever the four-coil configuration is used, which is not possible with three-coil WPT systems.

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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
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