Transmitting Capacity Estimation and Verification for Wireless Power Transmission System via Electromagnetic Resonant Coupling

(Tianjin Key Laboratory of Advanced Electrical Engineering and Energy Technology Tianjin Polytechnic University Tianjin 300387 China)



Transmitting Capacity Estimation and Verification for Wireless Power Transmission System via Electromagnetic Resonant Coupling

ZhangXianYuanZhaoyangZhangPengchengYangQingxinLiYang

(Tianjin Key Laboratory of Advanced Electrical Engineering and Energy Technology Tianjin Polytechnic University Tianjin 300387 China)

In this paper，the power distribution in the near-field of a magnetic dipole and the proportion its reactive power are obtained to estimate the optimum operating range of the wireless power transmission (WPT) system.Secondly，a general resonant coupling power transmission (RCPT) system is analyzed with thecoupled mode theory.The transferred power can be expressed asthe function of the quality factor，the coupling coefficient，and the angular frequency.For verification，a RCPT system composed of planar resonators is fabricated.It is shown by the measured results that，the maximum transmission distance is up to 5 m at about 10 MHz，which is an evident proof for the correctness of the proposed theory.

Resonant coupling power transmission，transmitting capacity，near-field estimation，coupled mode theory，planar resonator

0 Introduction

According to the frequency band and working principle of WPT，WPT technologies are generally classified as short-range，mid-range，and long-range applications.For short-range applications，the typical transmission distance ranges from a few millimeters to several centimeters by means of inductive coupling at tens of kHz.Mid-range applications are employed to transfer power in several meters at a few megahertz.In addition，there are also many literatures focusing on how power can be delivered over long distances in a range of several kilometers by using power laser or microwave transmission.Among them，RCPT is a method that based on strongly coupled magnetic resonances to transfer power wirelessly via non-radiative near-field between resonators.It is more convenient to charge the electronic products such as consumer electronics，and biomedical implanted devices，micro-sensors，etc[1-6].at wanted distances.

The capacity of power transmission distance is one of the key indicators of RCPT system.How to extend the power transmission distance should be considered first since the magnetic coupling decreases rapidly as the distance increases.In Ref.[7]，even and odd numbers of the repeaters are added in between the transmitter and receiver to maximize their output benefits.The transferred distance is about 80 cm which is 2.5 times the size of repeater.It should be noted that the misuse of repeaters would result in performance degradation which can be even worse than the repeater-free cases.In Ref.[8，9]，analytical and computational methods are used to identify conditions for wireless powering of sensors embedded in concrete when spaced 2.25 m.The transmission loss，propagation loss and the compound loss for normal incidence and oblique incidence is analyzed based on wave’s propagation model under the frequency of 0.1～1 GHz.However，it shows high sensitivity and poor operability for practical applications.In Ref.[10] an automated impedance matching system is proposed to increase the performance of system by matching the resonance frequency of the resonator connected to the power source.The transmitting distance is 30 cm which is the same to the size of the resonators.In Ref.[11，12]，60 W is delivered wirelessly by helical resonators with high quality factor when displaced 2 m at a frequency of 9.9 MHz.But there are no clear descriptions for the appropriate operating area.

In this paper，the capacity of transmission distance in RCPT system via non-radiative near-field is investigated. The power distribution of a magnetic dipole is achieved to determine the ideal theoretical operation area for RCPT system.Then by adopting coupled mode theory，the characteristic of power flow is expressed as a function of quality factor，coupling coefficient and angular frequency. And the contribution of each factor to the transferred power is discussed.A RCPT system composed of planar resonators is fabricated for validation.The light-emitting diodes are used as the load with low impedance for better distance extension.The measured results shows that the output power firstly increases and then decreases as the transmission distance increases from a very close distance to infinite，which is in consistent with the theoretical model.The transmission performance of repeater-free case is compared with the repeater assisted case.The maximum transmission distance is up to 5 m for an RCPT transmission system operates at about 10 MHz in experiment，which is in consistent with the proposed theory.

1 Power Distribution in the Adjacent Area of a Magnetic Dipole

The alternating power of electromagnetic emission sources (such as antenna) has different characteristics as the observation distance changes.In general，the electromagnetic field can be divided into near field and far field according to the distance between the observation point and the source.In the near field，there is no certain proportional relationship between the strength of magnetic field and electric field.In applications of antennas which has the characteristic of high voltage and low current，the electric field is much stronger than its magnetic field.For high current and low voltage applications such as induction heating applications，the magnetic field is much stronger than its electric field.The magnetic field density and electric field density decay very rapidly as distance increases.Also，the field distribution is uneven.However，in the far field cases where the field density is weak，the electromagnetic power is transferred by means of electromagnetic wave in which the field density decays much slower than in the cases of the near field.

As for RCPT systems，power flows between the transmitter and receiver through the near-field adjacent to the resonators.It has non-radiative features because the reactive power which comes in magnetic field and capacitance effect，is in a dominant position in the overall power constitution.The power circulates in the space adjacent to the power source without radiation in the near field which is known as the induction filed，thus making it a key prerequisite for RCPT systems that the transmission distance should not exceeds the near-field region in operation.However，the boundary between near-field and far-field is not fixed in space.Moreover，there are various models to describe different regions and boundaries between near-field and far field by many researchers as shown in Fig.1 whererindicates the distance from an observation point to the radiation source.

Fig.1 Two-and three-region models to describe the regions of near-field and far-field

As most models describe the boundaries between near-filed and far-field are given in electric dipole.The boundary conditions are functions ofr.In most researches in RCPT systems，power is delivered through magnetic field generated by resonators.Magnetic power is stressed as a key difference from inductive coupling.In order to determine the appropriate operation area for RCPT system，we investigate the power composition by elemental magnetic loop as shown in Fig.2，which is different from traditional models given by electric dipole.Through the integral of magnetic vector potential and duality principle of magnetic dipole，the field strength of the magnetic loop can be represented as Eq.(1)[13-15].

Fig.2 The magnetic loop located at the origin of the coordinate

In here，mis the instantaneous value of magnetic force；ωis the angular frequency per second；βis the wave vector which means the phase variation when the wave propagates in a unit length，orω/c，2π/λ；cis the speed of light，or 3×108m/s；ε0andμ0are permittivity and permeability of free space，respectively；η0is the free-space impedance，or 376.7 Ω；θis the angle between the zenith’s wire axis and the observation point；andΦis the angle betweenx-axis and the projection of observation vector on thex-yplane.

(1)

It can be observed from the equations that as distance increases，Eφdecreases.Eφreaches maximum whenr=0.ForHθ，the 1/rterm dominates the equations in the far-field.As the distance decreases，the 1/r3and 1/r2terms increase rapidly.Then，as a result，the 1/r3term dominates in the near-field.The distance changes would result in the variations inHoutcomes shown in Eq.(1) asrin here operates as an important and complex variation.As forHr，1/r3and 1/r2are the only two factors that matter.In this case，the 1/r3term dominates in the near-field and the 1/r2term dominates in the far-field.Examine the point at which 1/r3and 1/r2are equal which means the active power component equals to reactive power component，so this is the boundary condition between near-field and far-field as shown in Eq.(2).

(2)

(3)

Fig.3 The composition of S(er) as a function of βr

When this proportion turns out to be the lowest value in the graph，the power is mainly delivered in the form of active power，or in other words，nearly no reactive power is transferred here.That means，once the distance between resonators exceeds the near-field condition，the power delivered through magnetic field may be very limited or approximately 0 under certain distances which is not stable for real applications.Therefore，to operate in the region wherer<λ/2π is the first prerequisite when designing an RCPT system，because the reactive power would experience several cycles of variation whenrapproaches toλor more，even in some cases the reactive power can be approximately 0.In addition，the proportion of the reactive power in total power is very high within the region so more power can be delivered through magnetic field.

2 Capacity Estimation on Transmitting Distance Based on Coupled Mode Theory

It is clear to describe the general law of wave’s interaction by using coupled mode theory.For a general RCPT system composed of planar resonators as shown in Fig.4，the transmitting end contains source coilSand resonatora1while receiving end contains resonatora2and device coilD.The power onSorDis transferred or absorbed by induction of the resonators.The power on resonatorsa1anda2exchanges between each other through resonant coupling and can be described by an asymptotic model as Eq.(4)[16-18].

Fig.4 A general RCPT system composed ofplanar resonators

(4)

a1(t) anda2(t) denote the forward transfer mode of the resonators respectively；ω10andω20denote the natural resonant frequency of each resonator；κdenotes the mode coupling factor；Q1，2represent the quality factor of source coil and device coil respectively，which is often applied to illustrates the ability of power storage in non-linear circuit components.Γ1andΓ2denote the intrinsic loss of resonators，and there is a relationship between quality factorsQ1，2andΓ1，2thatΓ1，2=j/(2Q1，2).Meanwhile，ΓLis the intrinsic loss due to the load resistance，and there isΓL=j/(2QL) whereQLdenotes the quality factor of the load coil from the resonator side.Asejωtis a sinusoidal signal source with the amplitudeAsona1.

IfQ1andQ2are high enough，the first order Taylor expansion of the modified resonant angular frequencies can be expressed as Eq.(5).When there is certain loss on resonators，the variations of quality factor will bring shift to its resonant angular frequency in different degrees.

(5)

ω′10，20denote the modified angular frequency.

(6)

In order to extend the transmission distance，an ideal system is supposed to operate under critical coupling state in which there isω10=ω20=ω0.For a linear coupled system，considering the constraint of power conservation and comparing with circuit equations，there is

(7)

Wherek12denotes the coupling coefficient is betweena1anda2.When the system has a symmetrical structure，there isQ1=Q2=Q0.So the power transferred by RCPT can be derived by the two-norm form ofa2，as sh.0 own in Eq.(8).

(8)

Then，according to the principle of two-dimensional mean inequality，the maximum power，the receiving end obtain is Eq.(9).When conditionk12Q0=1 is met，receiving end gets its maximum power，which is also named critical coupling state.

(9)

To ensure a stable power transmission for a system，k12Q0≥1 is required.This is because the power would drop rapidly as distance increases when the condition is not met.Therefore，to analyze the system performance while considering parameters of the system under critical coupling，in other words，the required power on receiving end is less or equal to Eq.(9)，the power transmission capacity curve of a given RCPT system can be obtained whenk12Q0=1 in Fig.5，in wherek12，Q0andfare normalized and shown in longitudinal coordinates，respectively.

Fig.5 Power transmission capacity curve for RCPT system

In this system，the coil dimension，material and other physical parameters are fixed and changing the distance between transmitting coil and receiving coil.First，calculate the mutual inductance of coils by Neumann’s Formula as below.μ0is the vacuum permeability and dl1dl2means the unit differential element on the coils.ris the distance between coils andθis the angle between dl1and dl2.

(10)

Thenk12can be computed as

(11)

Since the material and geometry parameters of the system are fixed，the coupling coefficientk12can be calculated by both finite element method and analytical approach.ThenQ0，which is the precondition for critical coupling，can be derived according to the equation and is plotted in Fig.5.According to theQ0calculated above，the resonant frequency of the system can be derived by Eq.(5)，as the blue curve in the graph.In this way，the conditions which meet the requirements of critical coupling when the parameters of a RCPT system is fixed while changing the distance between resonant coils can be obtained.

It can be seen that there are always three intersections when drawing a line perpendicular to the horizontal axis.The physical significance of the three intersection points is that the minimum required quality factor and resonant frequency that enablea2to get enough power by RCPT when a given system operates under certain distance sincek12is fixed then.It should be noted thatQ0，fandk12are in an inverse relationship which means that the longer distance spaced between coils corresponds to smallerk12which requires greaterQ0and higherf.The given system gets its maximum power at the intersection of all three coils when the distance is approximately 0.9 m in this graph when all three curves intersect at one point.This is the best operation distance of a certain system which is related with the parameters listed above.When the system works within the distance of the intersection point，the system gets a high coupling coefficient andk12Q0>1 is a common case.So in this condition，the system works under the state of over coupling.The resonant frequency is always higher than the one shown in the graph.In this way，we do not need to concern too much about the conditions that coils should meet to enable stable power transmission.When the distance between the coils exceeds the intersection point in the graph，the system usually works under the state of under coupling for long transmission distance and low coupling coefficient.For these cases，resonant frequency and coupling coefficient should be higher than the values shown in the graph under certain distances to ensure a stable resonance.

In a word，from what has been mentioned above，we can look up for the maximum operation distance and the basic requirements for obtaining stable power transmission，especially in the cases of under coupling in the figure sinceQ0andfare determined.

3 Transmitting Capacity Verification on Planar Resonators via RCPT

Though the general trend of transmitted power can be derived by Eq.(8)，the quantitative measurement of the transmitted power is quite necessary.Power consumption comes from the resonator itself and load resistance.Quality factor implies the proportion of reactive power storage in circuit and power consumed in a cycle.The Hamiltonian form of driven power is

(12)

(13)

Fig.6 The active power on the load PL as a function of spaced distance

Fig.6 shows thatPLis not always increasing even thougha2is close enough toa1when driven frequency fixed.PLreaches the maximum value under a certain condition，then it gradually decreases regardless of distance variation.The maximum valuePLmaxand its necessary condition is shown as Eq.(14).

(14)

It is the optimum operating point of a RCPT system，because the active power on the load reaches the peak value with a distance long enough.Generally，the power transmission range is intended to be extended for RCPT to achieve more flexibility.When the distance is further expanded (i.e.k12continues to reduce)，PLdecreases rapidlyand tends to zero as shown in Fig.6.Becausek12is in a negative cubic order relation with space distance.

(15)

In order to verify the accuracy of the theory mentioned above，a RCPT system contains planar resonators is fabricated and its detailed parameters are shown in Table 1.The resonators have the same structure as the ones in Fig.4.They are tuned into the same resonant frequency which is 9.33 MHz in practical experiments.Sinusoidal signal is amplified by MOS tube BLF175 and the power is measured by power meter 4304A Bird Inc. and TPS 2014B Tektronix Inc.as shown in Fig.7a[19].

Tab.1 Parameters of RCPT system composed of plan a resonators

The device coil is connected to a 5 W LED so that the active power can be directly displayed by its brightness.

Because there is a small on-resistance and loss for LED，its impact on load quality factor can be reduced to the minimum.PLas a function of spaced distance is shown in Fig.6 where the optimum operation is at 1.25 m with a power of 4.87 W.When the distance is larger than 1.25 m，PLdecades rapidly to zero which is consistent with the results of Eq.(13) and Eq.(14).A group of signal LEDs has replaced the previous ones for the power is too small to light up the light.According to Eq.(2) and Eq.(3)，the near-field region at 9.33 MHz is about 5.12 m.In measurement，there is about 8～10 mW when spaced 4.98 m as shown in Fig.7b which basically confirmed the correctness of the model.In order to enhance the transmission performance，a repeater is adopted.In this case，20～35 mW is measured，as shown in Fig.7c.However，the near-field ofa1and repeater interacts with each other that it weakens the total effect which is the next step of the research.

Fig.7 Transmission capacity validation of RCPT system

4 Conclusion

Wireless power transmission via non-radiative near-field adjacent to the resonators is one of the significant features of RCPT.It is suggested that the power transmission should be operated within the near-field region in which the reactive power is in the dominant position.Its proportion in total power varies in period withβrand it is 91.93% whenr=λ/2π which can be considered as the boundary of the near-field and far-field.The power transmission capacity is mainly subjected tok12，Q0andfby coupled mode theory.A graph is plotted to illustrate the relationship betweenk12，Q0andfwith distance changes between coils whenk12Q0≥1.Detailed information of parameters can be observed in the graph to ensure a stable power transmission system under different distance conditions.The active power on the load is derived byQLas Eq.(13).It should be noted that it is not always increasing even thougha2is close enough toa1whenfis fixed.A maximum power peak exists as distance changes.In order to verify the accuracy of the theory mentioned above，a RCPT system contains planar resonators is fabricated that LEDs works as the load to decrease the fluctuation of load quality factor.When the system is driven at 9.33 MHz，the optimum transmission distance for the 5 W LED is at 1.25 m，with a power of 4.87 W.Moreover，the furthest transmission distance that can be measured and observed is 4.98 m for signal LED which is in consistent with the theoretical value of 5.12 m.

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Brief notes

Zhang Xian male，was born in 1983，vice professor.His research field contains numerical computing for engineering electromagnetic field and wireless power transmission technology.

Yuan Zhaoyang male，was born in 1991，Ph D，master student.His research field contains wireless power transmission technology.

基于电磁谐振耦合的无线电能传输系统传输能力估算与验证

张 献 苑朝阳 章鹏程 杨庆新 李 阳

(天津工业大学天津市电工电能新技术重点实验室 天津 300387)

首先研究了磁偶极子近场功率分布，并求得其中无功功率所占比例，以此为基础估算无线电能传输系统的最佳工作区域；其次，利用耦合模理论分析了典型谐振式无线传能系统，获得了以品质因数、耦合系数和角频率为变量的关于传输功率的函数；最后搭建了一套平板线圈谐振耦合无线传能样机，实验结果显示谐振频率为10 MHz时系统的最大工作距离为5 m，验证了所提出理论的正确性。

谐振耦合无线电能传输 传输能力 近场估算 耦合模 盘型谐振器

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