CHEN Xin ( )， HUANG Shoudao ()

1 College of Electrical and Information Engineering， Hunan University， Changsha 410082， China2 University of Humanities， Science and Technology， Loudi 417000， China

Abstract： The optimization method by adjusting load and distances between two adjacent coils (or resonators) is presented on basis of wireless power transfer (WPT) system with relay resonators. 2-port network and impedance matching theory are applied to analyzing power flow of incidence and reflection in WPT system， then setting up power flow model. The maximum power transmission efficiency can be obtained when the load and distance between secondary resonator and output coil meets impedance matching at 2-port network’s output port. The simulation and experimental results shown the impedance matching method can effectively improve and maintain transmission efficiency by adjusting load and distances between coils or relay resonators.

Key words： wireless power transfer; relay resonators; 2-port network; impedance matching; power flow model

Introduction

The technology and theory of wireless power transfer have emerged for a long time since the 19thcentury. A global system of wireless transmission of electricity power was proposed， but the Tesla experiment is very low in efficiency and considered dangerous. In the process of achieving this goal， obstacles include the low efficiency of transferring power over long distance. There are many approaches taken to improve it including coil design， topology and circuit optimization. Another approach to improve this phenomenon is using relay resonators.

The wireless power transfer (WPT)system with relay resonators has attracted substantial researchers and institutions to research and implement it. In 2006， the term WiTricity was led by Kursetal.[1]at Massachuetts Institute of Techology(MIT)， successfully demonstrating the ability of wirelessly light a 60 W light bulb. This WPT system has two copper resonator coils which similarly tuned resonant frequencies， and at only 40% efficiency[1]. The research project was spun off into a private company called WiTricity. WPT system with relay resonators is several times larger than traditional WPT system in the distance of transfer power. What’s more， it can also maintain high transmission efficiency if the WPT system is designed and controlled properly. Many researches have been showed in the fields of S-parameter[2-8]， coupled mode theory[1， 9-10]， 3-D finite element analysis (FEA)[11-13]and a frequency-tuning method[14-15].

In this paper，2-port network and impedance matching methods are used to design and optimize WPT system with relay resonators. Firstly， the paper analyzes the incidence and reflection of power flow through 2-port network and power wave theory， thus the power flow model is established. Secondly， impedance matching of the equivalent 2-port network with relay resonators is analyzed through power flow model. Lastly， the optimum design of WPT system with relay resonators is presented by optimizing the distance of two adjacent coils (resonators) as well as their load.

1 Port Network Analysis

As shown in Fig.1， the WPT system with relay resonators usually contains an AC power source， input coil， relay resonators， output coil and load. The distance between an input coil and a primary resonator isD12;D23is the distance between two resonators;D34is between a secondary resonator and an output coil.

Fig.1 Schematic diagram of WPT system with relay resonators

The primary coil is driven by an AC power source with high frequency and power move from the primary coil to the secondary coil through the air. The primary and secondary coils often connect capacity or other electronic components to realize circuit resonance and enhance its power transfer capability.

Fig.2 Block diagram of WPT system with S-S topology

In Fig.2VSis AC power source output voltage，RSis source impedance，L1/L2is Primary/Secondary coil inductance，Mis mutual inductance of coils，R1/R2is primary/secondary coil resistance，C1/C2is compensated capacitance of primary/secondary coil，RLis load.

The equivalent 2-port network of WPT system with S-S topology is shown in Fig.3. It includes coupling coils， resistance united to the 2-port network N2， the compensation capacitance of primary coil composed of the 2-port network N1and the compensation capacitance of secondary coil composed of the N32-port network.

Fig.3 The equivalent 2-port network with S-S topology

In Fig.3,Z01/Z02is input/output of characteristic impedance;V1is 2-port network input voltage;V2is 2-port network output voltage.

In order to facilitate analysis of 2-port network， it doesn’t consider the parasitic capacitance of coils.ωis the angular frequency in system operating， the transferABCD-parameter matrix of coupling coils is as follows.

(1)

Generally， operation frequency is less than 20 MHz and each element is connected through short-length lines， so it is unnecessary to consider the properties of circuit cable. As shown in Fig.3， the 2-port networks ofN1，N2， andN3are connected with cascade connection mode. Thus， theABCD-parameter matrix of the combined networkNof whole equivalent circuit is equal to the matrix multiplication of the three individualABCDparameter matrices.

(2)

Substituting compensated capacitanceC1/C2， coil resistanceR1/R2and equation to equation， theABCD-parameter matrices of 2-port networkNis obtained as

(3)

(4)

2 Power Flow Model

According toABCD-parameter matrix of 2-port network N， input active powerPIN， load powerPLand transmission efficiencyηare expressed as follows.

(5)

(6)

(7)

(8)

(9)

(10)

The characteristic impedance (Z01/Z02) is only related to the parameter (coil parameter and mutual inductance) of 2-port network. And when loadRLis equal to output characteristic impedanceZ02， input impedanceZinis exactly equal to input characteristic impedanceZ01.

In terms of theory of impedance matching of 2-port network and energy conservation law， Fig.4 analyzes power flow transfer direction， incidence and reflection and establishes the power flow model of WPT system.

Fig.4 Power flow model of WPT system

It considers the power flow when resonant frequency ofprimary side is equal to that of secondary side. In the primary side， source impedanceRSis not equal to 2-port network input impedanceZin(RS≠Zin) and source impedanceRSdoesn’t match for input characteristic impedanceZ01(RS≠Z01). Thus， power source output power is only part of the active power enter 2-port network. And because input impedanceZindoesn’t match input characteristic impedanceZ01(Zin≠Z01)， it leads to part of active power flowing into 2-port network reflect， namely the reflection power consumed in the input port through electromagnetic wave energy because of LC resonance. Meanwhile， small part of active power into the 2-port network is consumed by the resistance of input or output coil. In the secondary side， because loadRLis not equal to output characteristic impedanceZ02(RL≠Z02)， part of active power is reflected in 2-port network output port. This part of power is also consumed in the way of electromagnetic wave energy， andRLconsumes another part of power.

In terms of the power flow model and 2-port network theory， when the output port impedance is matched， there will be no power reflection at the input and output ports. Impedance matched of output port means that reflection coefficient ГLis equal to zero (RL=Z02) and the reflection coefficient Гinis also equal to zero (Zin=Z01). So transmission efficiency reaches maximum， and is simplified as follows.

(11)

The power flow model can effectively track power flow and power reflection by analyzing impedance matching of 2-port network and optimizing WPT system. When output characteristic impedanceZ02is unmatched for loadRL， it can reduce power reflection and improve transmission efficiency by adding impedance matching network or compensating network. Impedance matching network and compensating network in primary side play a role in regulating impedance. Thus， they can enhance or adjust the input power of 2-port network.

3 Modeling of WPT System with Relay Resonators

Figure 5 shows an equivalent circuit of WPT system with relay-resonators. The distance between the primary resonator and the secondary resonator (D23) is sufficiently large， and the input and output coils are usually small in WPT system with relay resonators. Thus， the mutual inductance between the input coil and the secondary resonator is as negligible as the one between the output coil and the primary resonator.

According to 2-port network theory and T-type coupling equivalent circuit of electromagnetic resonant coupling coils[8]， the model of WPT system with relay resonators can be simplified to an equivalent 2-port network in Fig.6. The input and output coils are respectively connected to AC power source and load.

Fig.5 Equivalent circuit of WPT system with relay-resonators

Fig.6 Equivalent 2-port network of WPT system with relay resonators

The equivalent 2-port network of WPT systemwith relay resonators can be divided into three 2-port networks:Nin，N， andNout.Nininput port is cascade-connected with the output port ofNout， and the output port ofNoutis cascade-connected with the input port ofNin. The 2-port networks ofNinandNoutplay a role in impedance matching.

(12)

(13)

(14)

(15)

In Eqs. (12) and (13)RL3=RL+R3，RS1=RS+R1.

The load power and the transmission efficiency of WPT system with relay resonators are related to the distance between coils (mutual inductance of coils)， loads and coil parameters. It is difficult to accurately calculate transmission efficiency and load power in Eqs.(12) and(13) because the above parameters are variable.

Characteristic impedanceZ01-relayandZ02-relayare only connected with the parameters of 2-port network without any relations to power source impedance and load. According to power flow model， transmission efficiency reaches to the maximum whenZ02-relayis equal to load. At the same time，Z01-relaydetermines active power input of 2-port network in a certain range， and ultimately determines load power.

4 Simulation and Experimental Verification

As shown in Fig.7， a WPT system with relay resonators is set up， the experimental platform includes DC voltage source (Xantrex XDC 10(V)-600(A))， power analyzer PZ4000， MOSFET driving module， LITZ coils， compensation capacitor and load，etc. As shown in Fig.8， input/output coils and relay resonators select double-coil of LITZ line. There are some coil parameters and electrical parameters of experimental platform in Table 1.

Fig.7 WPT system experiment platform

Fig.8 Coil and resonators photographs

ParameterValueLITZ line diameter/mm0.1×400Turns13Layer2Begin radius/mm30Terminal radius/mm70Coil inductance/μH76.1Coil resistance/Ω0.094Resonant frequency/kHz80Series compensation capacitor/nF52DC power source/V10DC voltage source resistance/Ω0.01

4.1 Simulation verification

The simulation calculation of LITZ coil inductance is 74.95 μH and its resistance is 0.1053 Ω using Maxwell finite element analysis. In Table 1， finite element simulation results are very close to measure results. Figure 9 is the variation curve of mutual inductance and distance between two coils with Maxwell simulation software and experimental data.

Fig.9 The distance between two coils versus mutual inductance

If experimental platform parameters in Table 1 are substituted into Eq.(15)， the three-dimensional waveforms of output characteristicZ02-relayvariations with mutual inductancesM23andM34when mutual inductanceM12at different constants is shown in Fig.10.

(a)

(b)

(a)M12=7.5 μH and (b)M12=15 μH

From Fig.10， output characteristic impedanceZ02-relaymainly depends on mutual inductanceM23andM34even if mutual inductanceM12slightly changes. Especially when mutual inductanceM34is low， the variation of mutual inductanceM23has little effect on output characteristic impedanceZ02-relay. Due to the symmetry of 2-port network， input characteristic impedanceZ01-relayis mainly determined by mutual inductanceM12.

To ensure that WPT system with relay resonators acquires higher transmission efficiency， the WPT system must meet output characteristic impedanceZ02-relayequal to loadRL. If mutual inductanceM34is low andZ02-relayand loadRLmatched， the change of mutual inductanceM23andM12should have little influence on transmission efficiency. This means slightly change of distanceD12andD23have little effect on the transmission efficiency.

Similarly， input characteristic impedance is mainly determined by distanceD12， the change of mutual inductanceM12causes significant changes of characteristic impedanceZ01-relay. So the variation of the distanceD12can directly adjust active power into the WPT system with relay resonators and eventually influencing load power. Meanwhile transmission efficiency remains relatively stable.

4.2 Experimental verification

Thewaveforms of transmission efficiency versus loadRLwith simulation and experiment at different distanceD34is shown in Fig. 11.

Fig.11 Transmission efficiency versus load (RL)

In this paper， transmission efficiency is the ratio of load powerand output active power of voltage source. Due to power loss of Metal-Qxide-Semiconductor Field-Effect Transistor(MOSFET) drive and compensation capacitors， transmission efficiency of experimental test is slightly lower than the results of formulation. The changes of distanceD12，D23andD34lead to output characteristic impedanceZ02-relayfluctuation， maximum transmission efficiency should be achieved at optimal load whenZ02-relayand LoadRLexactly matching. In Fig. 11， loadRLis equal to 11 Ω whenD12=50 mm，D23=70 mm andD34=50 mm， loadRLis equal to 5 Ω whenD12=50 mm，D23=70 mm andD34=70 mm.

Figure 12 shows the experimental and simulation curves of transmission efficiency versusD12，D23orD34whenD12=50 mm，D23=70 mm andD34=50 mm， respectively.

As shown in Fig. 10， the variation of mutual inductanceM12andM23have little influence on output characteristic

impedanceZ02-relaywhen mutual inductanceM34is low. So distanceD12and distanceD23have little effect on transmission efficiency when distanceD34is long enough and characteristic impedanceZ02-relayequals to loadRL. The transmission efficiency decreases rapidly when distanceD34increases， because output characteristic impedanceZ02-relayimpedance increase as distanceD34increases and it leads 2-port networkNoutput port impedance mismatched.

Figure 13 shows the curves of load power versus distanceD12whenD34=50 mm，D23=70 mm，RL=11 Ω andD23=70 mm，D34=70 mm，RL=5 Ω.

Similar with the impedanceZ02-relay， input characteristic impedanceZ01-relayis mainly determined by distanceD12. ImpedanceZ01-relaysignificantly decreases with distanceD12increases， and it causes more active power input to 2-port network increases， and obvious increase of load power， meanwhile transmission efficiency remains relatively stable.

Fig.12 Transmission efficiency versus D12， D23or D34

Fig.13 Load power versus distance D12

5 Conclusions

In this paper， WPT system with relay resonators is simplifiedto a 2-port network which is composed of two RLC 2-port network and a coupling coil. It analyzes how to optimize the load and the distance between two near coils(or resonators) and meet maximum transmission efficiency and load power regulation. To ensure WPT system with relay resonators realizes high transmission efficiency， impedanceZ02-relaymust achieve impedance matching with load. The impedancematching can be acquired by adjusting distanceD34or/and loadRL. The transmission efficiency can keep relatively stable as the slight variation of distanceD12andD23when impedanceZ02-relayand load matched. The variation of distanceD12leads to the change of impedanceZ01-relay， and can effectively improve/adjust load power. The optimal design method is proposed in this paper， which fully considers the relationship of the distance between coils and load. The change of the distance between coils(or resonators) can effectively improve the transmission efficiency of WPT system with relay resonators and realize the regulation of load power. These conclusions also can be extended to the WPT system with multi-resonators.