Study on Efficiency Maximization Design Principles for Wireless Power Transfer System Using Magnetic

Resonant Coupling

Hongchang Li, Xu Yang, Kangping Wang and Xiaoshuai Dong College of Electrical Engineering

Xi'an Jiaotong University Xi'an, China

Email: lhc@stu.xjtu.edu.cn

Abstract-Wireless Power Transfer (WPT) brings

convenience and safety in many applications and serves as a

research hot spot in recently years. Magnetic resonant coupling is

widely implemented in WPT applications such as mobile devices

and electric vehicles where large distance, large power amount

and high efficiency are the three key requirements in real

application. However, there are always trade-offs between these

requirements even in theory. Former literatures failed to

illustrate the complete relationships between these requirements.

Based on the phasor analysis, this paper illustrates the operating

principle of the whole circuit, attains the equivalent circuit

models of the system and derives the physical essence of

frequency characteristics. Then Maximum Efficiency Conditions

(MEC), which achieves the maximum efficiency without

sacrificing the requirements for power transfer distance and

power amount, is summarized. It should be noted that the highest

efficiency is achieved at the natural frequency of the receiver,

instead of any split frequencies of the coupled resonances.

Following the MEC, a WPT prototype was designed, which was

composed of a full bridge inverter, a LC resonant transmitter, a

LC resonant receiver and a full bridge rectifier. The resonant

frequency of the transmitter was designed to be slightly lower

than the inverter operation frequency-446kHz to make an

inductive load for the inverter so that all the MOSFETs operated

in Zero-Voltage-Switching (ZVS) condition. For the experimental

results, 300 W output power was obtained over a distance of

22cm with 84% overall efficiency.

Keywords-wireless power transfer; magnetic resonant coupling; maximum efficiency conditions

I. INTRO DUCTION

Wireless power transfer (WPT) was firstly investigated by Nikola Tesla in the early 20th century [1]. The advantages of transferring power wirelessly are significant due to the convenience and safety by removing the cables and manual plugs; batteries in some devices may be replaced by continuous wireless powering. Recently, WPT based on magnetic resonant coupling is demonstrated to be an efficient approach to transfer power in near-field [2]. Comparing with the other two popular WPT technologies - electromagnetic induction and the microwave power transfer, WPT using magnetic resonant

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coupling achieves both large power transfer distance and high efficiency at the same time.

The phenomenon of energy exchange between two coupled resonators is studies based on the coupled-mode theory in the beginning [3], latter the equivalent circuit model is more widely used [4][6][9][10]. It is found that the resonant frequencies of the resonant coupling system split into a lower frequency in odd mode and a higher frequency in even mode when the coupling is strong, even though the two natural frequencies of the independent resonators are identical [2][4]. And WPT system operates at the lower resonant frequency in odd mode is more efficient than that operates at the even mode [5]. It is also shown that within a certain region, although the coupling coefficient decays significantly when the power transfer distance increases, the efficiency keeps almost constant [6]. Based on these conclusions, frequency tracking methods for the WPT system are proposed to keep the system operating in the odd mode so that high efficiency power transfer can be achieved [7][8]. Moreover, some more complex resonant circuits are proposed to achieve more efficient power transfer by impedance matching [9] [10].

High efficiency is one of the most important objectives for WPT system design. In this paper, the Maximum Efficiency Conditions (MEC) of the WPT system using magnetic resonant coupling is summarized. Although it still focuses on the basic structure of the WPT system, the conclusions are suitable for the more complex systems. The operation principle and physical essence of frequency characteristics are illustrated based on the equivalent circuit model and phasor analysis. It shows that, the highest efficiency of power transfer is achieved at the natural frequency of the receiver, instead of any of the split resonant frequencies. Then, by defining the figure-of­merit of the circuit model, the coupling coefficient and load variations are investigated and the optimal load for a certain coupling is derived out. Finally, improving the quality factors of the transmitting and receiving resonances are illustrated to be the last step of efficiency maximization. According to the theoretical analysis, a high efficiency WPT system using magnetic resonant coupling is built to validate our method and conclusion.

II. ANAL Y S I S BA SE D ON THE EQU IVALENT C IRCU IT

MO DEL

A. Equivalent Circuit Model ojWPT Using Magnetic Resonant Coupling The circuit model with lumped parameters of the magnetic

resonant coupling is shown in Fig. 1. The only loss is caused by the equivalent series resistance (ESR) R] and R2 when the radiation is neglected. Based on the fundamental harmonic analysis (FHA), the feeding source is represented by an ideal sinusoidal voltage source Uim and the load with rectifier is replaced by an equivalent working resistance RIV•

Tn Fig. I, power is transferred using the mutual inductance in the near-field, which can be simplified as a quasi-static magnetic field when the radiation is neglected. At static state, the power transfer mechanism is described by 1· . .

UI = jmL1 .11 + jmM· 12

[/2 = jmM . II + jmL2 . 12

(1)

(2)

Where, UI and II are the phasors of voltage and current of L] respectively, similarly, U2 and 12 are the phasors of L2. The angular operation frequency is m, and j is the imaginary unit. The power transferred from the transmitter to receiver is represented by P. The phase difference between II and 12 is a. All these are shown in Fig. 2.

The expression of the transmitted power Pin (2) is matched with the result in [2], the only difference is the tenn sina, which is equal to 1 when a = 90°. The condition to make a = 90° is that the operation frequency is exactly equal to the natural frequency of the receiving resonator, i.e.

1 ())= --��C2

(3)

This can be explained by the decoupled equivalent circuits shown in Fig. 3. The input impedance of the receiver looking from the electromotive force is

(4)

And the reflected impedance in the transmitter is

(5)

When (3) is satisfied, Zin2 is resIstive and the phase difference between EM and 12 is 180°. In the other hand, EM leads 11 with 90°, therefore, the phase difference a between 11 and 12 is 90°.

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[]1 . � .CJ2

Uln Ll L2 Rw

Rl R2

Fig. I. Circuit model of magnetic resonant coupling.

(a) Coupled inductors.

U2 jwMl1

II

12

(b) Phasors. Fig. 2. Phasor analysis of the magnetic resonant coupling.

11 - - 12

� + L1 + � L2 + (, (2

Uin U, ZE EM U2

R1 R2

Fig. 3. The decoupled equivalent circuits.

Rw

As the only power loss is caused by the ESR, the input power, output power and efficiency can be defmed as following:

1] = � /l1n x 100%

B. The Phenomenon oj Frequency Splitting

(6)

(7)

(8)

When the resonant frequencies of the two independent resonances of transmitter and receiver are identical, i.e.:

1 1 -- = --- = 01 JLPI �L2C2 0

(9)

The resonant frequencies split from mo, if the resonators are coupled. This is described physically: when m<mo, Zin2 is capacitive so the reflected impedance Z" is inductive, while the series Lj and Cj branch is capacitive, therefore the inductive Z10

and the capacitive branch of L1 and C1 have an opportunity to make resistive input impedance Zinl looking from the input source. Similarly, the capacitive ZH and the inductive branch of LI and CI may make resistive input impedance Zinl when w>wo. The split frequencies satisfy the following equation:

Tm [JOJL] +_._1_+ Z1o ] = 0 JOJCI

(10)

The two split frequencies are denoted by Wodd «wo) and Weven (>wo) respectively. When the system runs at wodd, 12 leads EM with a phase angle between 0�90° because of the capacitive Zin2, and EM leads 11 with 90°, therefore the phase difference of 11 and h is 00<a<90°. The resonant currents are approximately in-phase. For this reason, running at Wodd is called "odd mode". Similarly, if running at Weven, the current phase difference is 900<a< 180°, the resonant currents are approximate Iy out -of­phase, so it is called "even mode".

When the resonant frequencies of the two independent resonances of transmitter and receiver are different, i.e.:

1 1 -- = � * -- = � ��C] �L2C2

(11)

It is proved that in this case, the resonant frequencies split from WI and W2 respectively. And wodd<min(w\,w2), weven>max( W\,W2)'

III. MAX IMUM EFF IC IENC Y CON D IT ION S

A. Operating at the natural frequency of the receiver To fmd the optimal operation frequency for high efficiency,

(8) is further derived as following

. R2 r·sma- --77 = ___ -----"OJ"-'CM,o::-

• 2 R] r·sma+ r . --OJM

Where r is the ratio of the magnitudes of 11 and h:

Considering the partial derivative

d77 1-77 dsin a . Rj sma+r-­

OJM

(12)

( 13)

(14)

Which is always above zero, therefore, the highest efficiency is achieved when sina = 1, i.e. a = 90° (all other variables are constant). The physical meaning is that, the natural frequency of the receiver should be equal to the operation frequency, in order to eliminate the reflected power from the receiver to transmitter.

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With the assumed parameters shown in Table 1, split frequencies and the corresponding efficiency are calculated and shown in Fig. 4, where the coupling coefficient is defined as

(15)

As can be seen, the efficiency at an operation frequency which equals to the natural frequency of the receiver is the highest. Fig. 5 shows the frequency characteristics when k is 0.1. Without loss of the generality, the natural frequencies are normalized to I, and there are no constrained units for parameters or variables.

TABLE!. ASSUMED PARAMTERS FOR SIMULATlON

Symbol Parameter Value L1• L2 Resonant inductance I Clo C2 Resonant capacitance 1

Rio R2 E SR 0.01

Rw Working resistance 0.1

Uin

� ----- odd

Input voltage 1

'0 1.4 ._.-iii _._._.. even ..................... ...

1.6

r I 5- 1 2 - ' ......... -.... , o� --<:�::�=:�:��������-��-�-�-�--�--��-o

0.1 0.2 0.3 0.4 0.5 Coupling eoeffieen! I , :v:

.... · ...... . . . - ---�--- =- od: w _._._.. even

-- 2 O L---�----�----��--------� o 0.1 0.2 0.3 0.4 0.5

Coupling eoeffieen!

Fig. 4. Split frequencies and efficiency.

Power and Efficiency 15 -----T-----�-----�------

� 10

[L 5

>­u c Q) :§ 0.4 w

1.1

-----T-----�----- �------

Q2 -----+----- 4-----�------

o �----�------�----�----� 0.9 0.95 1.05 1.1

Fig. 5. Frequency characteristics (k = 0.1).

B. Using the Optimal Load

When the operation frequency is already set at the natural frequency of the receiver, the resonant current ratio r should be optimized in order to minimize the ESR losses. This is achieved by the load matching. According to (12)

a1J = 0 � r = fK. �l + Jom2 + 1

ar vii; Jom

Where the figure-of-merit is defined as

mM Jom = --

�R1R2

(16)

( 17)

When both (9) and (16) are satisfied, the power transfer efficiency would be

The working resistance to satisfy ( 16) is

Rw = �1 + 10m2• R2

(18)

(19)

Therefore, to achieve the highest efficiency of power transfer, the load should be optimized according to (19).

C. Maximizing the Quality Factors a/the Transmitter and Receiver

When both condition A and B are satisfied, the only limit of improving efficiency is the figure-of-merit, shown in Fig. 6. Actually, the figure-of-merit can be transformed as

(20)

Where QJ and Q2 are quality factors of the transmitter and the receiver, respectively. As the coupling coefficient k is related to the power transfer distance, which is another specification of WPT system, k is usually limited by the sizes of devices and tends to be very low. Maximizing the quality factors QI and Q2 is the last possible method to improve the power transfer efficiency. And this requires better material and structure design of the resonators.

0.8

0.6

0.4

I I I I IIIII I I I I IIIII

I I I I IIIII _lJJ

I I I I I I I IIIII I I I I IIIII

I I I I IIII I I I I IIII

_1_1 J I I I I I I I IIII

1 1 11] I I I I IIII 11 1111- - T -II TlTII II IIII I I I I IIII II IIII I I I I IIII 0.2 -1 - r- H-t- I-t-II- - t- ---i ---t t-I+I

I I II IIII I I I I IIII II IIII I I I I IIII

o ---=���������� 10.1

10m Fig. 6. Efficiency vs. figure-of-merit.

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IV. EXPER IMENTAL RE SULT S

According to the theoretical analysis and efficiency maximization conditions, an experimental system was designed. Fig. 7 shows the structure of the system, which is composed of a full-bridge inverter, a transmitting LC series resonance, a receiving LC series resonance and a full-bridge rectifier. Parameters of the system are list in Table 2. These parameters were measured by the network analyzer and precision LCR meter.

Following the summarized efficiency maxImIzation conditions, the operation frequency (switching frequency of the inverter) was set to be equal to the resonant frequency of L2 and C2 which is 446 kHz. Fig. 8 shows that the 90° phase difference between the resonant currents at this condition. The load resistance was 21 Q and the equivalent working resistance approximately satisfies (19) therefore an optimal current magnitude ratio was also achieved. Moreover, in order to improve the quality factors, resonant coils were made by Litz wire and winded sparsely, shown in Fig. 9. It should be noted that the resonant frequency of the transmitter was lower than the operation frequency. This was intentionally designed to make an inductive load for the inverter so that all the MOSFETs operated in zero-voltage-switching (ZVS) mode and the switching losses was significantly reduced, shown in Fig. 10. With a 22 cm air gap, we got 300 W output power and the overall efficiency was 84%. This result validated our analysis and method.

+

Vin

M + ---RL Vo

Fig. 7. Structure of the experimental WPT system.

TABLE I I. PARAMTERS OF THE EXPERIMENTAL SYSTEM

Symbol Parameter Value LI Inductance of transmitter 133 IlH

CI Capacitance of transmitter 992 nF

Ji Resonant frequency of L, C1 438 kHz

QI Quality factor of transmitter 476

L2 Inductance of receiver 131 IlH

C2 Capacitance of receiver 974 nF

J2 Resonant frequency of L2 C, 446 kHz

Q2 Quality factor of receiver 458

k Coupling coefficient 0.048

j; Operation frequency 446 kHz

RL Load resistance 2 1[2

Fig. 8. Resonant currents.

Fig. 9. Transmitting and receiving coils.

Fig. 10. ZV S operation of the inverter.

V. CONCLU S ION

This paper analyzed the basic structure of WPT system using magnetic resonant coupling based on the lumped parameter circuit model and phasor analysis. The operation principles and frequency characteristics are illustrated in depth. With the theoretical analysis, MEC for WPT system is summarized. It was shown that the operation frequency should equal the resonant frequency of the receiver while the split frequencies were less important to achieve the highest efficiency. Optimal load was derived out using the definition of figure-of-merit. With the limit of coupling coefficient, the quality factors of transmitter and receiver were maximized to improve the efficiency. Experimental system was designed according to the analysis, and the result validated the summarized MEC. The overall efficiency of the system was 84% and the output power is 300 W over 22cm air gap.

The future work on this technology can be divided into two aspects: one is the study of the closed loop control, which keeps the system satisfying the maximum efficiency conditions when the distance or load is varying, and the other is improving

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the quality factors of the resonators in order to achieve higher power transfer efficiency.

REFERENCE S

[1] N. Tesla, U. S. patent 1,119,732 , 1914.

[2] Andre Kurs, Aristeidis Karalis, Robert Moffatt, 1. D. Joannopoulos, Peter Fisher, Marin SoljaCi6, "Wireless PowerTransfer via Strongly Coupled Magnetic Resonances," Science Express, vol. 317. no. 5834, pp. 83-86, June 2007.

[3] Herma A. Haus and Weiping Huang, "Coupled-Mode Theory," Proceeding of the IEEE, vol. 19, No. 10, Oct. 1991.

[4] Takehiro Imura, Hiroyuki Okabe and Yoichi Hori, "Basic Experimental Study on Helical Antennas of Wireless Power Transfer for Electric Vehicles by using Magnetic Resonant Couplings," Vehicle Power and Propulsion Conference, 2009. IEEE Pages 936-940.

[5] Y. Kim and H. Ling, "Investigation of coupled mode behaviour of electrically small meander antennas," Electronic Letters, vol. 43, No. 23, 8th November 2007.

[6] Alanson P. Sample, David A. Meyer and Joshua R. Smith, "Analysis, Experimental Results, and Range Adaptation of Magnetically Coupled Resonators for Wireless Power Transfer," IEEE Trans. Ind. Electron., vol. 58, No. 2, pp.544-554, Febrary 2011.

[7] Jongmin Park, Youndo Tak, Yoongoo Kim, Youngwook Kim, and Sangwook Nam, "Investigation of Adaptive Matching Methods for Near-Field Wireless Power Transfer," IEEE Trans. Antennas and Propagation, vol. 59, No. 5, May 2011.

[8] N. Y. Kim, K. Y. Kim, 1. Choi and c.-w. Kim, "Adaptive frequency with power-level tracking system for efficient magnetic resonance wireless power transfer," Electronics Letters, vol. 48, No. 8, 12th April 2012.

[9] Zhen Ning Low, Raul Andres Chinga and Jenshan Lin, " Design and Test of a High-Power High-Efficiency Loosely Coupled Planar Wireless Power Transfer System," IEEE Trans. Ind. Electron., vol. 56, No. 5, May 2009.

[10] S. H. Lee and R. D. Lorenz, "Development and validation of model for 95% efficiency, 220 W wireless power transfer over a 30cm air-gap," in Proc. Energy Conversion Congress and Exposition, Atlanta, pp. 885-892, Sep. 201O.