# Efficiency Analysis of Multiple-Transmitter Wireless Power Transfer Systems

Authors: Jin Zhang; Fangfang Wang
Journal: International Journal of Antennas and Propagation
 (2018)

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## Abstract

The range of the power transmission region can be effectively enlarged by a multiple-transmitter (TXs) and single-receiver (RX) magnetic wireless power transfer (WPT) system. However, the power transfer efficiency (PTE) of the WPT system with two TXs and a single RX has been intensively studied in the past few years. Therefore, a deep analysis of multiple-TX WPT systems needs to be implemented urgently to achieve the highest PTE in a wider range of power transmission. This paper analyzes a general multiple-TX single-RX system in detail based on the circuit model. Two optimal conditions, in which the feeding amplitude ratios need be equal to the mutual inductance ratios and the optimal load should be connected to RX, are derived to obtain the highest PTE. In the experiment part, the feeding amplitude ratios for multiple TXs are implemented by a proposed lumped transformer for the first time. The use of the lumped transformer reduces the complexity compared with the use of multiple inverters to achieve predetermined feeding amplitude ratios. Multiple-TX systems are designed and measured to validate the theoretical analysis. Experimental results agree very well with theoretical calculations.


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## Body

## 1. Introduction

The research of magnetic-resonance coupling is particularly prized in the areas of wireless power transfer (WPT) since this research was published in the US journal Science in 2007 [1]. Many interesting fields on magnetic-resonance WPT have been studied in just the last 10 years alone [2–19].In [2, 3], the circuit model and evaluation method for conventional and multiple-coil WPT systems are presented to discuss the frequency splitting properties. The frequency splitting results in degraded output power at the original frequency when the transfer distance is shortened. To deal with the power transfer degradation due to frequency splitting, the schemes of nonidentical transmitting and receiving coils, antiparallel resonant loop transmitter (TX), and mixed magnetic and electric coupling have been proposed to prevent frequency splitting via suppressing the coupling strength in references [4–6]. Methods of Kirchhoff’s voltage law, band-pass filter theory, and coupled-mode theory are compared and discussed in [7, 8]. Based on those methods, characteristic analysis of goal setting WPT systems are implemented in [9–19]. The systems consisting of multiple receivers (RXs) can be applied well to the scenario of a single charging platform. This scenario supplies power for various electronic devices; therefore, multiple-RX systems have acquired comprehensive and deep studies [13, 15, 16]. On the other hand, the multiple-TX cases are considered for adding gain over longer distances comparing with single-TX cases [14, 17, 18]. As the number of TXs increases, the gain of the system is enhanced over a larger region. In [14], the authors presented a two-TX system which is optimized by the distance between the two TXs. The investigation shows that the proposed two-TX system can generate a more stable output power in a broader space than the single-TX case. Reference [17] introduced a practical power line synchronization technique with regard to frequency and phase to synchronize all TXs. In [18], since two TXs and a single RX are located at an arbitrary position and rotation angle, the amplitude ratio and the phase difference of the signals of the two TXs are optimized to obtain the highest power transfer efficiency (PTE). However, in these studies [14, 17, 18], the analysis model and available conclusions are limited to two-TX systems.In this paper, the PTE of a multiple-TX WPT system is investigated. The analytical expression of PTE is derived. We observe that the relationship between the amplitude in each TX and the mutual inductance between the RX and any TX crucially determines the highest PTE. Based on the analysis, two experimental schemes of efficiencies versus the number of TXs and power transfer distance are implemented later in this paper.


## 2. Theoretical Derivation of Transmission Efficiency

The operating wavelengths are considerably larger than the sizes of power coils and the transmission distance. Thus, circuit theory can be used to analyze the model of the WPT system. Figure1 illustrates the equivalent circuit of a representative WPT system with multiple TXs and a single RX. TXii=1,2,…,n is the expression of the ith TX. The mutual inductance between TXi and RX is represented as MiR. LR, CR, and rR are the self-inductance, compensation capacitor, and equivalent series resistance (ESR), respectively, of the RX. Similarly, Li, Ci, and ri are the self-inductance, compensation capacitor, and ESR, respectively, of the ith TX. The self-resonant frequency for each resonator is ω0 (ω0=1/LiCi=1/LrCr).Figure 1
Circuit model of a multiple-TX single-RX WPT system.Based on Kirchhoff’s voltage law (KVL), at operating frequencyω, the electrical relationship of the multiple-TX WPT system shown in Figure 1 can be described as
(1)V1V2⋮Vn0=Z1jωM12⋯jωM1njωM1RjωM12Z2⋯jωM2njωM2R⋮⋮⋮⋱⋮jωM1njωM2n⋯ZnjωMnRjωM1RjωM2R⋯jωMnRZR×I1I2⋮InIR,where Vi and Iii=1,2,…,n are peak-value phasors, and all of Vi are in phase. Zi=ri+jωLi−1/ωCi and ZR=rR+RL+jωLR−1/ωCR are the impedances of each TX and the single RX, respectively. In general, compared to the mutual inductance strengths between TXs and the RX (MiR), the ones between TXs (Mij,i,j=1,2,…,n, and i≠j) are small enough to be omitted, namely, Mij=0. Therefore, under the resonance condition ωLi−1/ωCi∣ω=ω0=0 and Mij=0, formula (1) can be simplified to
(2)V1V2⋮Vn0=r10⋯0jω0M1R0r2⋯0jω0M2R⋮⋮⋮⋱⋮00⋯rnjω0MnRjω0M1Rjω0M2R⋯jω0MnRrR+RL×I1I2⋮InIR.Mesh currentsIi and IR in TXi and RX can be merged into two sets of (3) and (4) from (2):
(3)Vi=riIi+jω0MiRIR,(4)0=∑i=1njω0MiRIi+rR+RLIR.Solving (3), the formula for Ii in terms of IR is
(5)Ii=Vi−jω0MiRIRri.Substituting (5) into (4) gets
(6)IR=−∑i=1njω0MiRVi/ri∑i=1nω0MiR2/ri+rR+RL.Solving simultaneous equations of (3) and (6), we obtain
(7)IiIR=−Viri×∑t=1nω0MtR2/rt+rR+RL∑t=1njω0MtRVt/rt−jω0MiRri.The PTE for the system is(8)PTE=RLIR2∑i=1nRiIi2+rR+RLIR2=RL∑i=1nRiIi/IR2+rR+RL,where Ii and IR are the currents flowing through the ith TX and the load without considering radiation loss. Substituting (7) into (8), PTE can be represented as
(9)PTE=RL∑i=1nrirR+RLVi/ri+ω02∑t≠inMtRMtRVi−MiRVt/rirt2/ω0∑i=1nMiRVi/ri2+rR+RL.It can be found from (9) that PTE achieves the optimal value, PTEOPT′, with fixed RL when MtRVi−MiRVt=0, namely,
(10)ViVt=MiRMtR.Under condition of (10), we obtain
(11)PTEOPT′=RLω02∑i=1nMiR2/rirR+RLrR+RL+ω02∑i=1nMiR2/ri=ω02∑i=1nMiR2/riRL+rRrR+ω02∑i=1nMiR2/ri/RL+2rR+ω02∑i=1nMiR2/ri.From (11), the optimal load, RL,OPT, can be derived to obtain the final optimal efficiency, PTEOPT, when RL=rRrR+ω02∑i=1nMiR2/ri/RL.
(12)RL,OPT=rR1+ω02rR∑i=1nMiR2ri,(13)PTEOPT=1+ω02/rR∑i=1nMiR2/ri−11+ω02/rR∑i=1nMiR2/ri+1.We set(14)An=1+ω02rR∑i=1nMiR2ri.The expressions of optimal load and transfer efficiency in (12) and (13) can be rewritten as
(15)RL,OPT=rRAn,(16)PTEOPT=RL,OPT−rRRL,OPT+rR=An−1An+1.Forms of expression in (15) and (16) for calculating the optimal load and transfer efficiency for the multiple-TX WPT system are identical to the forms for multiple-RX systems presented in reference [15]. It can also be found that both RL,OPT and PTEOPT increase with the number of TXs. The reason for this is that the total coupling between the TX and RX increases and the equivalent parallel ESR reduces when the number of TXs increases.


## 3. Numerical Calculations and Experiments

In order to illustrate the analyses in part II, we use MATLAB software to perform the numerical calculations. An important target of using such multiple-TX WPT techniques is to improve the energy efficiency of the system. Thus, in the designed system, multiple small TXs with the same size and a large RX are considered, which is used for decreasing the coupling between TXs to satisfy the theoretical presuppositions. To facilitate comparison discussion for the multiple-TX WPT system, coils of TXs are placed side to side with a coil of RX in this part for numerical calculations and experiments. The center of the study area,o, is set as the origin of a rectangular coordinate, which is shown with dashed red line in Figure 2. The RX loaded with resistance moves in the study area to investigate the PTE. The center of the RX does not go out of the boundary of the study area. Namely, point o′ always moves within the grey dotted circle, shown as Figure 2.Figure 2
The RX moving within the circle of the study area for optimal PTE investigation.The radii of RX and the study area areRRX and RSAi, respectively, where i=1,2,…,n for various numbers of TXs. The multiple TXs are distributed evenly around the study area. The system layouts for two, three, and four TXs are shown in Figures 3(a)–3(c), respectively. The radius of each TX is RTX, and the distance between TXs is D for each case. Therefore, the radii of the study area for two-, three-, and four-TX cases are RSA1=D/2−RTX+RRX, RSA2=D/3−RTX+RRX and RSA3=D/2−RTX+RRX, respectively. It should be clear that RSA1 < RSA2 < RSA3. Table 1 provides a summary of relevant dimensions for the system.Figure 3
Spatial layout for a multiple-TX single-RX system: (a) two TX, (b) three TX, and (c) four TX.
(a)
(b)
(c)Table 1
Dimensions of system structure.
Dimension
Value
D
1 m
RTX
5.5 cm
RRX
15.75 cm
RSA1
29 cm
RSA2
36.9 cm
RSA3
49.5 cm
Turn number ofTX
25
Turn number ofRX
11
High ofTX
3.4 cm
High ofRX
1.5 cmThe general calculation method of mutual inductance between each TX and RX has been presented in the former works of our team [2, 8]. The electrical parameters of TX and RX are listed as follows: parasitic resistances including the ESR of external series capacitor, self-inductance, and capacitance of external series capacitor of TX are 5 Ω, 198.6 μH, and 110.8 pF, respectively, and the counterparts of RX are 5.1 Ω, 132.8 μH, and 165.7 pF, respectively. The capacitance values of capacitors are selected to make the TXs and RX resonate at the same frequency of 1.073 MHz.Figure4(a) shows the efficiency in the circular study area under a fixed voltage ratio and load, V1:V2=1:5 and RL=10Ω. It can be found find that the three maximum efficiency regions occur around TX2, which is fed by a high-voltage amplitude. It is noticed that the efficiency increases nonmonotonously with a decreasing distance between RX and TX2. PTEOPT′ obtained under conditions of V1:V2=M1R:M2R and RL=10Ω is shown in Figure 4(b). Comparing with the efficiency without any optimal operation shown in Figure 4(a), the efficiency in this case is obviously higher, and maximum values are close to TXs. Under conditions of V1:V2=M1R:M2R and RL=RL,OPT shown in Figure 4(d), the optimal efficiency of PTEOPT is plotted in Figure 4(c). It can be observed that the efficiency is further improved comparing with Figure 4(b).Figure 4
Calculated values of a two-TX system when RX is moving in the study area. Efficiency distribution in the study area (a) without optimization, (b) under conditions ofV1:V2=M1R:M2R and RL=10Ω, and (c) under conditions of V1:V2=M1R:M2R and RL=RL,OPT. (d) Optimal load for PTEOPT.
(a)
(b)
(c)
(d)Loading withRL=10Ω, the nonoptimized efficiencies for the three-TX system under V1:V2:V3=1:5:10 and the four-TX case under V1:V2:V3:V4=1:5:10:20 are presented in Figure 5(a) and Figure 6(a), respectively. The maximum efficiency appears close to the TX fed with the highest voltage amplitude and towards the TX with the second-highest voltage amplitude, such as the position of (0.058, 0.33) for maximum PTE=0.26 close to TX3 and towards TX2 in the three-TX system. Similarly, the position of (−0.25, 0.43) for maximum PTE=0.24 is close to TX4 and towards TX3 and (0.23, 0.39) for the secondary maximum PTE=0.04 is close to TX3 and towards TX4 in the four-TX case.Figure 5
Calculated values of a three-TX system when RX is moving in the study area. Efficiency distribution in the study area (a) without optimization, (b) under conditions ofV1:V2:V3=M1R:M2R:M3R and RL=10Ω, and (c) under conditions of V1:V2:V3=M1R:M2R:M3R and RL=RL,OPT. (d) Optimal load for PTEOPT.
(a)
(b)
(c)
(d)Figure 6
Calculated values of a four-TX system when RX is moving in the study area. Efficiency distribution in the study area (a) without optimization, (b) under conditions ofV1:V2:V3:V4=M1R:M2R:M3R:M4R and RL=10Ω, and (c) under conditions of V1:V2:V3:V4=M1R:M2R:M3R:M4R and RL=RL,OPT. (d) Optimal load for PTEOPT.
(a)
(b)
(c)
(d)In the condition ofV1:V2:V3=M1R:M2R:M3R, Figure 5(b) plots the optimal efficiency PTEOPT′ under RL=10Ω and Figure 5(c) plots the final optimal efficiency PTEOPT under the optimal load RL,OPT shown in Figure 5(d) in the three-TX system. Similarly, in the four-TX case, when V1:V2:V3:V4=M1R:M2R:M3R:M4R, Figure 6(b) plots PTEOPT′ under RL=10Ω and Figure 6(c) plots PTEOPT under RL=RL,OPT shown in Figure 6(d). As a result, PTEOPT > PTEOPT′ > PTE in each case when RX moves in the circle of the study area. All of the maximum PTEOPTs of two-, three-, and four-TX systems are 0.79 when the RX is RTX + RRX apart from any one of the TXs. It shows that the maximum efficiency mainly depends on the mutual inductance between the nearest TX and the RX when the system satisfies the two optimization conditions. Comparing with the multiple-TX systems, we can find that the effective charging areas increase with the increasing number of TXs. Thus, using a multiple-TX system can maximize the flexibility of the WPT application.From the investigations of the optimal PTE in Figures4, 5, and 6 for two-, three-, and four-TX systems, respectively, two problems need to be illuminated:
(1)
The maximum values of optimal PTE in the three kinds of multiple-TX systems are the same result of 0.791 and always appear at the closest position to the TXs. Similarly, the optimal load values for the maximum PTE are also the same. These are due to the distance between TXsD=1 m, which is large enough that the mutual coupling effect of the nearest TX is much higher than those of the other TXs. The mutual inductance between RX and the nearest TX is almost five times the inductance between RX and the other TXs.
(2)
Two parameters of PTE and power delivered to the load (PDL) are the main criterions to evaluate the performance of a WPT system. The high PTE means a low system loss; however, if feeding power is largely limited under optimal PTE conditions, the PDL of this system would be very low. Here, we present the PDL of a multiple-TX system under the optimal PTE conditions. The specific values of PDL can be calculated with specific values of feeding voltage. Therefore, setting up reference values ofM0R=10−5H and V0=1V, Vi,PTEi=1,…,n can be obtained according to V0:V1:⋯:Vn=M0R:M1R:⋯:MnR and the calculated mutual inductance value of MjRj=0,…,n for a different receive location of RX. The calculated feeding voltage values of TX1 for two-, three-, and four-TX systems versus the position of RX are shown in Figure 7. The other feeding voltage values of TXkk=2,…,n are circular symmetric with those of TX1 about the center of the study area. Using the feeding voltage in Figure 7 and optimal load in Figures 4(d), 5(d), and 6(d), namely, under optimal PTE conditions, Figure 8 gives the PDL for two-, three-, and four-TX systems. It can be found that the maximum PDL is 0.26 W using maximum feeding voltage value of 0.74 V. This power level of PDL is sufficient for the charging of consumer electronics and the nodes in internet of things, and so on, through proper adjustment of feeding voltage amplitudes. Therefore, the studies in this paper are significant. Similarly, the maximum feeding voltage and PDL for two-, three-, and four-TX systems are the same due to the long distances between TXs selected in this paper.Figure 7
The voltage values ofTX1 for calculating PDL under conditions of maximizing PTE: (a) two-, (b) three-, and (c) four-TX systems.
(a)
(b)
(c)Figure 8
The values of PDL under conditions of maximizing PTE: (a) two-, (b) three-, and (c) four-TX systems.
(a)
(b)
(c)According to the theoretical analysis presenting in Section2, we manufacture five TXs and one RX with the sizes shown in the numerical calculation part to measure the PTEOPT. To ensure each TX with the predetermined amplitude and phase of feeding voltage, a lumped transformer with a Mn-Zn-type toroid ferrite core is used to deliver the power of one channel coming from the source into several channels feeding to multiple TXs. Each TX is set at the same distance away from the RX, and then the amplitudes of the feeding voltage for at most five TXs are set to identical values to obtain optimal efficiency. As shown in the subgraph in Figure 9(a), the lumped transformer consists of six groups of winding on a ferrite core for a five-TX system. The turn number of each group of winding is the same, 3 for this experiment, and one group of winding connects the function generator and the others feed the identical power to the multiple TXs. Figure 9(a) shows the overall experiment setup for the five-TX case.Figure 9
Experimental investigation ofPTEOPT versus the number of TXs: (a) example of experimental setup; (b) optimal load for PTEOPT and measured values of PTEOPT comparing with the calculated data versus the number of TXs.
(a)
(b)Figure9(b) shows the calculated and experimental values of PTEOPT and the optimal load for PTEOPT versus the number of TXs when DiR=0.26mi=1,2,…,5, corresponding to MiR=2.17μH, where DiR is the distance between the RX and each one of the TXs. The calculated PTEOPT and RL,OPT increases with increasing number of TXs, which is consistent with the analysis in Section 2. The experimental results of PTEOPT′ of at most a five-TX system shown in circular dots match well with the calculated results.To further confirm the conclusion obtained from analysis, two TXs and one RX are arranged face to face to construct a complete system, and the distance between the two TXs is fixed at 0.6 m. Using the dual channel function generator shown in Figure9(a), the amplitude ratio and phase difference of two feeding voltages can be adjusted to arbitrary values. For comparison, two PTEs of voltage ratios of V1:V2=1:1 and 1 : 3 with RL=50Ω are investigated. Calculated and measured efficiencies with and without the optimal conditions are compared in Figure 10(b) when the distance between RX and TX1, D1R, changes from 0.05 m to 0.55 m. The optimal mutual inductance ratios and loads for PTEOPT and PTEOPT′ are plotted in Figure 10(a). It can be found that PTE=PTEOPT′ at D1R=0.3m and 0.37 m for V1:V2=1:1 and 1 : 3, respectively. The reason for these is that M1R:M2R=1 and 0.33 as shown in Figure 10(a), which satisfies the optimal condition of V1:V2=M1R:M2R to maximize the PTE with an unoptimized load. As a result, the PTEOPT shown in black line in Figure 10(b) under the conditions of V1:V2=M1R:M2R and RL=RL,OPT has the highest values. Duo to the influence of a strong paramagnetic response [2, 19], the measured values for the scenario of the fixed voltage ratio and load slightly deviate from the calculated values when RX gets close to TX. But overall, the measurements for different scenarios agree with the calculated values.Figure 10
Efficiencies versus transfer distance when two TXs and a single RX are arranged face to face to construct a complete system: (a) mutual inductance ratio and optimal load for the highest efficiency; (b) calculated and measured efficiency values under different conditions.
(a)
(b)In this section, we also present the PTE characteristics of a one-TX and one-RX system. The values of PTE of a one-TX system with an optimal load and a fixed load of50Ω are plotted in Figure 10. It is clear that the optimal PTE of the two-TX system is slightly larger than that of the one-TX system when RX is close to TX1 since M1R2/r1 for the one-TX system is almost equal to ∑i=12MiR2/ri for the two-TX system in this case; however, when RX is close to TX2, M1R2/r1 is much less than ∑i=12MiR2/ri which results in the value of PTE for the one-TX system being far below that for the two-TX system.


## 4. Conclusions

In this paper, the procedures of efficiencies for a multiple-TX WPT system are analyzed in details based on the circuit model. There are two steps to obtain the highest efficiency via the theoretical analysis. Firstly, the feeding voltage ratio should be set to the mutual inductance ratio to achieve a higher PTE. Then the optimal load needs be attached to the RX to obtain the highest efficiency. The expressions for the highest efficiency and optimal loads for any multiple-TX system can be unified to the same form. Numerical results of two-, three-, and four-TX systems are further illustrated to compare the optimal and nonoptimized efficiencies. Experiments for a two-TX system are carried out to verify the numerical analysis when transmission distance is changed. Theoretical and experimental results indicate that the optimal efficiency increases along with the number of TXs mounting up.In terms of two optimal conditions for the highest efficiency, the expressions of the conditions are simple but universal, which provide a guideline for the optimal design of a multiple-TX WPT system.The proposed multiple-TX system is easy to fit on the surrounding environment (such as four TXs embedded in the four corners of a table and one RX positioned on any point of the table) to achieve a large effective charging area.



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