A Generalized Methodology to Generate, Amplify and Compensate Multi-Frequency Power for a Single-Inverter-Based MF-MR-S-WPT System

To meet the unique challenges on how to generate, amplify and compensate the multi-frequency (MF) power in MF multi-receiver simultaneous wireless power transfer (MF-MR-S-WPT) systems, this article proposes an innovative and generalized methodology. In proposed methodology, a MF modulation method based on cosine switching frequency modulation (CSFM) technique and a general multi-resonant transmitting tank design method based on circuit synthesis theory are introduced. With the proposed MF modulation method, a standard full-bridge inverter can be used to generate MF power, leading to a simple configuration of transmitting source. Moreover, a modulation degree of freedom can be provided to change the power transfer ratio, achieving power redistribution among receivers though software implementation. With the proposed design method of multi-resonant transmitting tank, the use of complicated multiple harmonic analysis can be avoided and the arbitrary multiple power components at selected resonant frequencies can be effectively amplified and completely compensated. Finally, the effectiveness of proposed methodology is verified theoretically and experimentally.


I. INTRODUCTION
An unique feature of magnetically-coupled wireless power transfer (WPT) system is the capability of using a single transmitter to simultaneously charge multiple receivers, such as electric vehicles [1] and portable devices [2], [3]. According to whether the resonant frequencies of receivers are designed to be identical or different, the multiple-receiver simultaneous WPT (MR-S-WPT) systems can be classified into two categories: single-frequency MR-S-WPT (SF-MR-S-WPT) system [4]- [10] and multi-frequency MR-S-WPT (MF-MR-S-WPT) system [11]- [19]. In SF-MR-S-WPT system, the receivers have same resonant frequency as that of the transmitter and consequently the power transferred to different receivers is must though a common power channel. It means that the influence of cross couplings among receivers is difficult to be reduced and the power redistribution among The associate editor coordinating the review of this manuscript and approving it for publication was S. Ali Arefifar . receivers cannot be achieved by just using the transmitter [9], [10]. On the other hand, more attentions have been paid to the MF-MR-S-WPT system. In this system, the receivers are designed with different resonant frequencies. In the meantime, the transmitter provides the power of different frequencies. By tuning the resonant frequencies of receivers to one of the frequencies emitted from transmitter, the separate power channels can be established in MF-MR-S-WPT system, which would be very helpful to reduce the cross coupling influence and achieve power redistribution by just using the transmitter. However, MF-MR-S-WPT system presents unique challenges, especially on how to generate, amplify and compensate the multi-frequency (MF) power.
For generating MF power in MF-MR-S-WPT system, various inverter-based methods have been proposed. According to the number of inverters being used, these methods can be classified into two types: multi-inverter-based method [11]- [13] and single-inverter-based method [14]- [19]. In multi-inverter-based method, the multiple inverters are VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ operating at different switching frequencies and can be connected to individual transformers [11], [12] or multiple transmitting coils [13] to combine MF power from transmitter to receivers. Based on this method, the power of different receivers can be controlled independently by changing the duty cycles of corresponding inverters [12]. However, the use of multiple inverters may increase system complexity and generate unexpected power losses. Instead, a single inverter can be used to generate MF power and even achieve power redistribution among receivers, leading to a relatively simple system configuration. One way of generating MF power with a single inverter is to use a modified configuration of full-bridge inverter [14], [15]. In [14] and [15], one more diode is connected in series to each inverter leg to synthesize a half-cycle sinusoidal current by applying the superposition methodology of fundamental and oddharmonic components. In comparison with a standard fullbridge inverter, the additional diodes will increase the system loss and complexity. In addition to the modified configuration, the MF modulation methods have been presented to generate MF power for a standard full-bridge inverter [16]- [19]. The simplest MF modulation method of standard full-bridge inverter is to generate a square-waveform output voltage and the fundamental and harmonic components of this voltage can be used to simultaneously charge multiple receivers of different resonant frequencies [16], [17]. However, its disadvantages is the unchangeable component amplitude, which fails in adjusting power transfer ratio to receivers. Another MF modulation method of a standard full-bridge inverter is based on the selective harmonic elimination (SHE) technique, namely multi-frequency programmed pulse-width modulation (MFPWM) [18], [19]. In this method, the modulated full-bridge inverter can generate a dual-frequency output of 100kHz and 6.78MHz or frequencies within ranges of 87-300kHz. However, a large computational burden arises in this modulation method and therefore makes it difficult to be extended to the applications with more than two receivers. Likewise, this MF modulation method is also not a generalized one for a standard full-bridge inverter. Except the challenge on how to generate MF power in MF-MR-S-WPT system, the amplification and compensation of MF power components are two important subjects to be further considered and their solutions are mainly based on the design of transmitting tank. Most of the configurations of transmitting tanks in the existing MF-MR-S-WPT systems are using a single series capacitor [4], [8], [11], [13], [20], [21]. This configuration is widely adopted in the single-transmitter single-receiver WPT system [22], [23], but it does not suitable to MF-MR-S-WPT system because only one resonant frequency can be provided at a time. As a result, only the selective receiver with same resonant frequency can get power and thus a simultaneous WPT cannot be achieved. To provide more resonant frequencies, a multi-resonant transmitting tank composed of multiple LC topologies is designed in [12], [16] and [17]. The multi-resonant tank is able to extract and amplify the power of different frequencies. Further, in [16] and [17], the reactive component at each resonant frequency is eliminated by adding a compensation tank between the multi-resonant tank and transmitting coil. However, in [12], [16] and [17], only the design process of multi-resonant transmitting tank for the case with two receivers is introduced in detail and their design methods are complicated because the multiple harmonic analysis is required. Hence, there is still a lack of simple and general design method of multi-resonant transmitting tank which can amplify and compensate the MF power components for applications with arbitrary multiple receivers.
This article demonstrates an innovative and generalized methodology to generate, amplify and compensate the MF power for a single-inverter-based MF-MR-S-WPT system, which could be easily extended to the applications with arbitrary multiple receivers. The proposed methodology has two parts: MF modulation method for a standard full-bridge and generalized design method for multi-resonant transmitting tank. The proposed MF modulation method is based on the cosine switching frequency modulation (CSFM) technique and can generate the MF power with adjustable component amplitudes. It means that a modulation degree of freedom is provided to change the power transfer ratio to receivers. The proposed general design method of multi-resonant transmitting tank is based on circuit synthesis theory and therefore can avoid the use of complicated multiple harmonic analysis. With designed tank, the MF power can be amplified to enhance power transfer level and the reactive components at each resonant frequency can be eliminated to increase system efficiency. Finally, the proposed methodology is evaluated theoretica-lly and experimentally though a sample with three receivers.
The rest of this article is organized as follows. The proposed MF-MR-S-WPT system is described in Section II. The CSFM-based MF modulation method for a standard full-bridge inverter is introduced in Section III. In Section IV, the circuit-synthesis-based general design method for multi-resonant transmitting tank is presented. The theoretical analysis of the modulated and designed system is given in Section V and experimental results are shown in Section VI. Finally, the conclusions of this article are given in Section VII. Fig. 1 shows the circuit topology of the proposed MF-MR-S-WPT system. The system uses a standard full-bridge inverter to generate a mixed-frequency driving voltage v d , which can provide power of different frequencies. Multiple receivers are designed at different resonant frequencies and each of them is tuned to one of the frequencies emitted by the MF transmitter. Consequently, the power of different frequencies from transmitter can be simultaneously transferred to multiple receivers though individual power channels of specific frequencies. In proposed system, the power transfer level of each channel can be enhanced by using a multi-resonant tank in the transmitting side. Meanwhile, the phase between driving voltage v d and current i d at each resonant frequency is controlled by inserting a MF compensation tank between multi-resonant tank and transmitting coil. As shown in Fig. 1, the chosen multi-resonant tank is a combination of n LC topologies, known as Cauer network, to provide n resonant frequencies [24]. The chosen MF compensation tank is a series combination of one capacitor C f0 with several parallel LC topologies, known as Foster network [25]. To independently control the input phase at each resonant frequency and be guaranteed to find the positive values of Foster network parameters, the number of parallel LC topologies in Foster network should be n-1 for the system with n receivers.

II. THE PROPOSED MF-MR-S-WPT SYSTEM
In Fig.1, V in and i in are the input dc voltage and input current from dc source. L ci and C ci are the inductance and capacitance of the ith (i = 1, . . . , n) LC topology in Cauer network. The inductance and capacitance of the ith LC topology in Foster network are noted as L fi and C fi , respectively. L t and R t are the self-inductance and equivalent series resistance of transmitting coil. L ri and R r i is the self-inductance and equivalent series resistance of the ith receiving coil. M ti is the mutual inductance between the transmitting coil and the ith receiving coil. M ij (i = 1, . . . , n; j = 1, . . . , n; i = j) are the mutual inductances among receiving coils. i t and i ri are currents of transmitting coil and the ith receiving coil, respectively. C ri is the series compensation capacitance of the ith receiving coil. In the ith receiver, the dc output voltage V o i is obtained by using an uncontrollable half-bridge rectifier connecting to a filter capacitor C di and feeding a passive load R Li . S i (i = 1, 2, 3, 4) stands for the switching signals of full-bridge inverter.

III. THE CSFM-BASED MF MODULATION METHOD
The first objective of this article is to generate the MF power by just using a single inverter. This objective can be achieved by using the MF modulation method for an inverter. As well known, the simplest single-inverter-based MF modulation method is to generate a square-waveform output voltage v d . This MF modulation method is called the fixed switching frequency modulation (FSFM) in this article. Based on the Fourier series analysis, the v d in FSFM can be given by where A = 4V in /π and ω c = 2πf c . f c is the fixed switching frequency of inverter, which is also called the center frequency.
The frequency components of square-waveform voltage v d , such as fundamental and third-order harmonics, can be used to simultaneously charge two receivers of different resonant frequencies [16], [17]. However, FSFM method has two drawbacks: unchangeable harmonic amplitude and large harmonic frequency difference. The former drawback may result in a nonadjustable power transfer ratio to receivers and the latter one will cause a seriously unbalanced loss distribution among receivers. To avoid these two drawbacks, a CSFM-based MF modulation method has been proposed in this article. CSFM is a kind of periodic switching frequency modulation technique and widely applied for pulsewidth-modulated converters to reduce EMI [26]. This article has extended this modulation method to the MF-MR-S-WPT system.

A. BASIC SCHEME OF CSFM
In CSFM, the switching frequency of full-bridge inverter is changed in a cosine manner as where f m is the modulation frequency and K m is the modulation index. When compared with FSFM, the spectrum of driving voltage v d in CSFM will spread into a wider frequency range with reduced harmonic amplitudes. It means each original harmonic of v d in FSFM will be modulated by CSFM into more components of different frequencies. By using CSFM, the kth modulated harmonic of v d is given by where θ k is the phase angle of the kth modulated harmonic. In CFSM, θ k is obtained by where ω m = 2πf m . m f = 2πK m /ω m is the modulation degree of freedom which can be used to change the component amplitudes for adjusting power transfer ratio to receivers. VOLUME 8, 2020 Consequently, the driving voltage v d in CSFM can be expressed by According to (A1) and (A2) in Appendix-A, v CFSM d showing its frequency components is derived in (6), as shown at the bottom of the next page. In (6), J n (n = 0, 1, . . .) is the nth-order Bessel function.
It can be noted in (6) that each original harmonic of v d in FSFM is modulated by CSFM into more components with equally spaced frequencies. As clearly shown in (6), the multiple frequencies can be maintained by using the proposed CSFM-based MF modulation method. More importantly, the amplitude of each frequency component can be adjusted by changing the modulation degree of freedom m f . It means that, based on CSFM, it is able to change power transfer ratio to receivers by just using a single inverter. Among the frequency components, those derived from the modulated fundamental F CFSM 1 are selected in this article because F CFSM 1 has a much higher amplitude than other modulated harmonics F CFSM k (k = 3, 5, . . .). Thus, the specific frequencies of individual power channels for n receivers, namely selected resonant frequencies, are chosen among the frequencies derived from F CFSM 1 by where f i is the ith selected resonant frequency.    , as illustrated in Fig. 3, where m f is set to 1.0 and f c and f m are set to 100kHz and 80kHz. It can be noted in Fig. 3 that the frequencies and duty cycles of pulses of v d are different but v d still varies in a periodic manner. The switching signals of full-bridge inverter can be generated by comparing the remainders of θ 1 (t)/2π with π, as shown in Fig. 4.

IV. DESIGN METHODOLOGY OF COMPOSITE TRANSMITTING TANK
The second objective of proposed methodology is to amplify and compensate the MF power components. This objective can be achieved by using a composite transmitting tank, as shown in Fig. 1. The proposed composite transmitting tank is composed of multi-resonant tank and MF compensation tank. The multi-resonant tank and MF compensation tank are used to amplify and compensate the MF power components, respectively. Although the literatures [12], [16], and [17] have introduced the design method of multi-resonant tank, they require the complicated multiple harmonic analysis and does not have generality. The proposed designed method is based on circuit synthesis theory, leading to a relatively easy and more generalized task.

A. DESIGN OF MULTI-RESONANT TANK
The proposed multi-resonant tank uses the Cauer network. Fig. 5 shows the circuit model of n-order Cauer network in complex-frequency domain. The Cauer network exhibits relatively low sensitivity to nonexact values of its components, which is an important feature for WPT design. The n-order Cauer network can provide n resonant frequencies and its voltage transfer function in general form can be expresses by where V d and V co are the input and output voltages of Cauer network. H 0 is a constant. ω i is the ith natural resonant angular frequency of Cauer network, at which V co reach its peak value. To amplify the MF power components of selected resonant frequencies f i , the natural resonant angular frequency ω i of Cauer network should be chosen by Besides, to establish separate power channels, the resonant frequency of each receiver should be designed to one of selected resonant frequencies by tuning its series compensation capacitance C r i as C ri = 1 (4π 2 f ri L ri )and f ri = f i (10) where f r i is the resonant frequency of the ith receiver. The proposed design method of Cauer network is based on circuit synthesis of voltage transfer function H (s), which can avoid the use of complicated multiple harmonic analysis. The Cauer network is a two-port LC network and its Z -parameter equation in complex-frequency domain is given by For I co = 0, according to (11), the voltage transfer function H (s) can be expressed by using Z parameters as It should be noted in (12) that the poles of H (s) should be the zeros of Z 11 (s). Therefore, the circuit synthesis of H (s) is to find the input impendence Z 11 (s) of Cauer network. In this article, Z 11 (s) for n-order Cauer network should be selected as where ω p i(i = 1, . . . , n − 1) is set to 0.5(ω i + ω i+1 ). B is the magnification coefficient. By using polynomial division, the Z 11 (s) in (13) can be transformed into where b 1 ∼ b 2 n are the constant coefficients which equal to the values of Cauer network components. In other words, L c1 , C c1 , L c2 , C c2 , . . . , L cn , C cn in n-order Cauer network can be obtained by calculating b 1 ∼ b 2 n as   (15). Here, three resonant frequencies are chosen to f 1 = 100kHz, f 2 = 180kHz, and f 3 = 260kHz, respectively. The magnification coefficient B in (13) is chosen to 2/π/f 1 . As shown in Fig. 6, the designed Cauer network can amplify its output voltage at each selected resonant frequency and further establish separate power channels with an enhanced power transfer level.
where D and A 1 are the determinants of coefficient matrix and can be calculated in (A5) and (A6), respectively. According to circuit theory, the input impendence Z in of system can be obtained by where T ij (i = 1, 2 and j = 1, 2) is the T parameter of Cauer network, which can be calculated in (A7). The expected equivalent reactance X * f of Foster network, at which the imaginary part of Z in becomes zero, can be yielded by solving (18).
However, it is difficult to get the analytical solutions because the order of (18) is high. Therefore, instead of the analytical solutions, the numerical solutions of (18) can be used to find X * f . For the MF-MR-S-WPT system with n receivers, n different selected resonant frequencies f 1 , f 2 , . . . , f n are required and can be chosen in (7). To independently control the input phase at each selected resonant frequency, the Forster network needs to provide n different values of X * f responding to n selected resonant frequencies. That is, X * f (ω 1 ), X * f (ω 2 ), . . . , X * f (ω n ). For n-order Foster network shown in Fig. 7, its equivalent reactance X f is expressed as It can be noted that in (19) that the Foster network has the feasibility of compensating both inductive and capacitive reactive components because the equivalent reactance ωL fi /(1-ω 2 L fi C fi ) of one parallel LC topology in Foster network can be designed to be positive or negative value. That is why the Foster network (a series combination of one capacitor with several parallel LC topologies) is chosen as the MF compensation tank in this article because it has generality for compensating reactive component of any value. As noted in (19), the adding of one parallel LC topology in Foster network has the potential to provide one more different value of X f . To provide n different values of X f for independent control of input phase at all selected resonant frequencies and be guaranteed to find the positive value of Foster network parameters, the number of parallel LC topologies in Foster network should be n-1 for the system with n receivers. The component parameters of Foster network can be determined in the following steps.
Step1: Determine the denominator 1-ω 2 L fi C fi in (19) by choosing the product term L fi C fi to be a certain value. In this article, L fi C fi is chosen to p ω 2 i . ω i is the ith selected resonant angular frequency, given in (9). p is a constant. For example, p can be chosen as 0.95. If the calculated component parameters of Foster network are negative, p is changed to be more than 1. Here, p can be chosen as 1.05.
Step2: Substitute L fi C fi = p ω 2 i to (19) and establish the following linear equation considering X * f at n selected resonant frequencies.
where X is the n × n coefficient matrix and can be expressed by The C f0 and L f1 to L fn can be calculated by . . .
Step 3: Calculate the rest of components C f1 to C fn by 181518 VOLUME 8, 2020  (7), which are emitted from MF transmitter. And then, the resonant frequencies of receivers are designed to the selected resonant frequencies by tuning the series compensation capacitances C r i in (10). Next, the power channels to couple transmitter with receivers though multi-resonant tank are established by choosing the natural resonant frequencies of Cauer network to the selected resonant frequencies. Based on the circuit synthesis of input impedance Z 11 , the components of Cauer network can be found in (15). Lastly, according to the expected equivalent reactance of Foster network at each selected resonant frequency, the components of Foster network can be found in (22) and (23).

V. THEORETICAL ANALYSIS OF PROPOSED SYSTEM A. INPUT IMPENDENCE
For simplifying the theoretical analysis, the proposed system with three receivers is considered as a representative example and its equivalent circuit at any single one angular frequency ω is given in Fig. 9, where the input impendence Z (ω) in without using the designed transmitting tank, with just using the designed multi-resonant tank, and with simultaneously using the designed multi-resonant tank and MF compensation tank are calculated, respectively. Fig. 10 shows the calculated angles of Z (ω) in at different frequencies and the specifications are listed as follows: Then, according to the design flow in Fig. 8, the parameters of designed transmitting tank are calculated and listed as follows: 74nF, L f1 = 6.24µH, C f1 = 103.57nF, L f2 = 2.19µH, and C f2 = 211.13nF. As noted in Fig. 10, by using the designed MF compensation tank, the angles of Z (ω) in at each selected resonant frequency become zero. It means that the MF reactive components through separate power channels are completely eliminated.
As shown in Fig. 10, although the MF compensation tank is designed to eliminate the reactive components at selected resonant frequencies, it fails in controlling the phase between driving voltage v d and current i d to zero at all frequencies, such as sideband frequencies f c -f m , |f c -2f m |, and |f c -3f m |, and FIGURE 9. The equivalent circuit of proposed system with three receivers at any single one angular frequency ω. VOLUME 8, 2020 so on. As a result, the zero-current switching of inverter cannot be maintained all the time. Nevertheless, switching losses caused by non-selected frequency components are small because the majority of power transferred to receivers is through individual power channels of selected resonant frequencies. For this reason, the reactive power at non-selected frequencies is also small.

B. OUTPUT POWER
By using the Thevenin's theorem and considering the total equivalent impedance, the circuit form in Fig. 9 changes from the left-side compound circuit to the right-side simple one. As previously addressed, X (ω) f represents the equivalent reactance of MF compensation tank.V (ω) s and X (ω) s represent the Thevenin equivalent voltage source and impedance, respectively, looking from the output terminals of multi-resonant tank.
By applying Cramer's rule, three receiving coil current phasorsİ where With using the designed transmitting tank, the output power of the kth receiver at any single one angular frequency ω is obtained by According to (6), by using CSFM, the absolution value of driving voltage phasor at the ith selected resonant frequency can be given by Substituting (28) into (27), the output power of the kth receiver at the ith selected resonant frequency is given by Based on the multi-frequency multi-magnitude superposition methodology of power components at selected resonant frequencies, the total output power of the kth receiver by applying the CSFM can be given by By tuning the resonant frequencies of receivers to one of the frequencies emitted from transmitter, the separate power channels are established in MF-MR-S-WPT system. Consequently, the vast majority of power transferred to receivers is through individual power channels of selected resonant frequencies and therefore the total output power of receiver is approximated to the output power only at itself resonant frequency. So, the expression of output power of receiver in (30) can be simplified. Moreover, it can be noted in (29) that by using the proposed CSFM, the output power of receivers can be adjusted by changing the modulation degree of freedom m f . Fig. 11 shows the calculated total and approximate output power of three receivers at different values of m f , with and without using the designed transmitting tank. For the case without using the designed transmitting tank, the output power can be calculated similarly by using (29) and (30), but here D (ω i ) s in (29) should be replaced by D (ω i ) which is given in (A5) and T (ω i ) 11 in (29) should be set to be one instead. As shown in Fig. 11 (a) and (b), regardless of whether it uses or does not use the designed transmitting tank, the curves of total and approximate output power of receivers have a good coincidence. Nevertheless, comparing Fig. 11 (b) with Fig. 11 (a), the power transfer level is enhanced greatly with using the designed multi-resonant tank. It means that the MF power through separate power channels is amplified.

VI. EXPERIMENTAL RESULTS
To verify the validity of proposed system and corresponding MF modulation method and general design method, a setup of the system with three receivers is built in the laboratory, as shown in Fig. 12. The proposed CSFM  method for full-bridge inverters is implemented using DSP TMS320F28335. The waveforms and data are recorded by the oscilloscope (Agilent DSO 5034A). The parameters of the system are measured by using an LCR meter (HIOKI 3532-50) and listed in Table 1. The system will be tested in two steps. In the first step, to clearly demonstrate the effectiveness of power amplification and reactive compensation with using the designed transmitting tank, the system is tested at single selected resonant frequency. Here, the driving voltage is sinusoidal and can be provided through the function generator (Tektronix AFG3022) and power amplifier (NF HSA4012). Then, the system is tested at all selected resonant frequencies to verify the MF power generation and power redistribution among receivers. Instead, the transmitting composite compensation is connected to a full-bridge inverter which is modulated by using CSFM to provided a MF driving voltage with adjustable component amplitudes. The input voltage of full-bridge inverter is generated by a dc supply (IT6723H).  Fig. 13 shows the experimental waveforms of the proposed system tested at single selected resonant frequency, where three resonant frequencies for the system with three receivers are chosen to f 1 = 100kHz, f 2 = 180kHz, and f 3 = 260kHz, respectively. As shown in Fig. 13 (a), (c), and (e), the phase angles between driving voltage v d and driving current i d at each selected resonant frequency are compensated to be very close to zero by using the designed MF compensation tank. It can be noted in Fig. 13 (b), (d) and (f) that the output voltage of the receiver at itself selected resonant frequency is much higher than other two output voltages. That means the designed multi-resonant tank can extract and amplify the MF power at selected resonant frequencies and then establish the separate power channels. As observed in Fig. 13 (e) and (f), a small driving voltage at the third resonant frequency can be used to generated a relatively large output voltage. This is because the voltage gain of the design multi-resonant tank at that frequency is much higher than others at different selected resonant frequencies, as shown in Fig. 6.

A. VERIFICATION OF MF POWER AMPLIFICATION AND COMPENSATION
It can be noted in Fig. 13 that when the system is operating at single selected resonant frequency, only the receiver which has same resonant frequency can get power and others can hardly be charged. It means that the influence of the cross coupling among receivers is less. There are two reasons. One is the receivers in experiments are placed without overlapping and therefore the mutual inductances among receiving coils are much smaller than that between transmitting coil and receiving coils. The other reason is that by introducing the multi-frequency concept to establish separate power channels, the cross coupling influence is inherently reduced.   set to be zero, CSFM becomes FSFM. As analyzed in (6), the original harmonic of v d in FSFM is modulated by CSFM into more components with equally spaced frequencies and it can choose the value of m f to change the amplitudes of frequency components. Fig. 15 shows the experimental waveforms of the proposed system tested by using CSFM at two different values of m f . The pulses of driving voltage v d in CSFM have a variable frequency and duty cycle, which excite the MF driving current i d as well as MF transmitting current i t . When m f is 0.5, as shown in Fig. 14 (b), the frequency component of v d at first selected resonant frequency f 1 = f c is much higher than those at other two selected resonant frequencies f 2 = f c + f m and f 3 = f c +2f m . Therefore, the output voltage V o1 of first receiver is highest, as shown in Fig. 15 (b). By changing m f from 0.5 into 1, as shown in Fig. 14 (c), the frequency component of v d at f 1 is reduced while other two selected frequency components at f 2 and f 3 are increased. It brings a reduction in V o1 and the increase in V o2 and V o3 . Consequently, the power transfer ratio to receivers is changed.

C. VERIFICATION OF POWER RESDISTRIBUTION
Further experimental results of the power transferred to different receivers along with the system efficiency are shown in Fig. 16, where the value of m f is changed from 0 to 1.5. As noted in (30), the total output power of three receivers is approximated to the output power at individual resonant frequencies f 1 = f c , f 2 = f c + f m , and f 3 = f c +2f m , which are proportional to [J 0 (m f )] 2 , [J 1 (m f )] 2 and [J 2 (m f )] 2 , respectively. As the increase of m f in the range between 0 and 1.5, J 0 (m f ) is reduced while J 1 (m f ) and J 2 (m f ) are increased. Therefore, as shown in Fig. 16, the power P o1 transferred to  the first receiver becomes smaller and the power P o2 and P o3 transferred to the second and last receivers become larger. It can be noted in Fig. 16 that P o3 increases quickly to a large value after m f is bigger than 1. As previously addressed, this is because the voltage gain of the designed multi-resonant tank at f 3 is much higher than others at f 1 and f 2 . It suggests that m f should not be chosen too large.
This article focuses on the low power applications, such as mobile phones and swarm robots, where the spatial freedom is tight. Therefore, the size of coils being used in experiments is small and consequently the measured power is not very large. Nevertheless, it does not limit the application of propo-sed system forward to a higher power level. As shown in Fig. 16, just by using the proposed single-inverter-based MF modulation method, any two of receivers can pick up same power and work at the rated power level at the same time. However, since only one modulation degree of freedom is provided in proposed MF modulation method, it would not suitable for more receivers unless the multiple receiving coils are designed with different sizes. The proposed method is particularly suitable to charge the devices of different power levels in a same zone.

VII. CONCLUSION
In this article, a novel and generalized methodology to generate, amplify and compensate the MF power for a singleinverter-based MF-MR-S-WPT system is proposed. The main advantages of proposed methodology include: 1) Based on the proposed CSFM method, a standard full-bridge inverter can be used to generate MF power, leading to a simple configuration of transmitting (Tx) source. 2) With CSFM, a modulation degree of freedom can be provided to change the power transfer ratio, achieving power redistribution among receivers though software implementation. 3) The proposed general design method of multi-resonant transmitting tank is based on circuit synthesis theory and therefore can avoid the use of complicated multiple harmonic analysis. 4) The proposed methodology could be easily extended to the applications with arbitrary multiple receivers. To highlight these advantages, Table 2 shows some comparisons between the proposed and existing MF-MR-S-WPT systems in [11]- [19]. It can be noted that the system that uses the proposed methodology is obviously better than the existing MF-MR-S-WPT systems. Unfortunately, the individual power control cannot be achieved in this article, which will be studied in the future.  whereV t andİ t are the excitation voltage and current phasors of transmitting coil.İ ri (i = 1, . . . , n) is the current phasor of the ith receiving coil. Z t = R t +j ωL t . Z r i = R r i + R reci +jωL r i + 1/(jωC r i) and R reci = 4R Li /π 2 is the equivalent input resistance of the ith uncontrollable half-bridge rectifier. By applying Cramer's rule, the solution of I t can be obtained as where D, as given in (A5), is the determinant of n × n coefficient matrix of (A3) and A 1 , as given in (A6), is the determinant of that coefficient matrix with its first row and first column removed. For n-order Cauer network shown in Fig. 5, its T -parameter matrix can be obtained as where T i (i = 1, . . . , n) is the T -parameter matrix of the ith LC circuit in Cauer network and can be calculated by