Efficient Wireless Power Transfer ResonanceDoes Not Imply High Efficiency

Kazuya Yamaguchi, University of Miyazaki Ichijo Hodaka, University of Miyazaki

AbstractThis paper naively inquires when power transmission isoptimized by tuning possible parameters of a wireless power transfersystem. It is well known that tuning the frequency of the AC supplyinput implies maximizing the magnitude of transmitted power intoa receiver side. This type of tuned wireless power transfer system,however, may cause a low efficiency of power transfer. The paperillustrates that such an undesired situation exists, and suggests thattuning parameters in the both views of large magnitude of transmittedpower and high efficient transmission will be desirable for betterwireless power transfer systems.

Keywordswireless power transfer, highly efficient power trans-mission, resonance

I. INTRODUCTION

MOST of devices that are driven by electric powerare usually fed via electric wires connected to ACpower supply. On the other hand, some of electric devicesare required or designed to be fed without electric wires in the way of called WPT (wireless power transfer). Sincethe amount of power provided by WPT is usually restrictedcompared with wired power transfer, WPT has not been usedfor many applications.

In 2007, a successful experiment that could wirelessly trans-fer practical amount of power away from sixty centimeterswas reported[1]. The heart of the experiment was to generateresonant phenomena of electromagnetic coupling betweentransmitting and receiving coils. After the pioneering study[1],many researchers have tried to develop theory and experimentsfor WPT. In [2] a structure of circuit was devised in order toraise the factor Q of circuits, since better WPT needs highervalues of the factor Q. In [3] devised antennas are proposed tobe used for a directed energy radiation in order to implementefficient WPT. In [4] the effect of radiation energy to humanbody was discussed.

It is generally understood that resonant phenomena will becaused if one tunes the frequency of AC power supply tothe natural frequency of a circuit. The natural frequency isdetermined by the values of circuit elements; especially, themutual inductances which relate transmitting and receivingsides of the circuit play an important role. These valuesare determined by the radii, winding numbers, and relativeposition of two coils[5]. It is important to know the relationbetween the relative position of coils and mutual inductancesbecause the position of the coils could be changed in variousreasons.

For extending the usage of WPT, the most important spec-ification for WPT is power and efficiency of transmission. Inmany literatures, causing resonant phenomena should lead tohigh power and efficient WPT. However, little is known about

how resonance brings the best performance of WPT based ona concrete and mathematical aspects.

In this paper, we propose a series of procedures to analyzerelations between the resonant frequency, obtained averagepower at the receiving side, and the ratio of the average powerat the receiving side to the average power at the transmittingside that is called an efficiency of WPT. Our proceduresare based on the differential equations in the form whichcommonly used in control theory. This gives us a benefit thatwe can manipulate different transfer functions and write therelations under investigation in a clear and unified manner.Consequently, we will assert that resonance does not implythe optimization of efficiency of WPT in general, by observingderived equations.

To illustrate this situation that should be carefully treated,we pick one of most common circuits for WPT, and putpractical values into the circuit elements, and then we demon-strate the situation numerically. In fact, although we havethe case when resonance is equivalent to maximization ofefficiency of WPT, we have another case when resonance leadsto loss of efficiency of WPT. Thus we propose to use the wayof modelling by mathematical equations and to describe thetargets into equations for WPT systems.

II. ANALYSIS OF WIRELESS POWER TRANSFER CIRCUIT

A. Self inductance and mutual inductanceWe consider the situation that current i flows in a coil with

a radius r1 and a winding number n1, called coil 1. Thisgenerates magnetic flux in the whole space by the Biot-Savartlaw. Here we consider a magnetic flux density B which isaway from the current i by z and which is along the centralaxis of the coil 1[6]. That is,

B =i

4

2n1r21

(z2 + r21)3=2

: (1)

Where is the permeability.Now we consider another coil 2 with a radius r2 and a

winding number n2. From equation (1), self inductances L1of the coil 1 and L2 of the coil 2, and mutual inductances M1and M2 between the coils 1 and 2 can be written as below.

L1 =n21r1

2; L2 =

n22r2

2

M1 =n1n2r

21r

22

2(z2 + r21)3=2

; M2 =n1n2r

21r

22

2(z2 + r22)3=2

:(2)

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From the above equations, note that these mutual induc-tances are inversely proportional to cube of the distance zbetween two coils[7].

B. Wireless power transfer circuit and its mathematical modelWe study a typical circuit for WPT depicted below[8].

Despite of the placement of two coils in the figure, we assumethey have a common central axis as explained in the previoussection.

Figure 1 A Wirelss Power Transfer Circuit

The resistor R1 is supposed to represent an inter-nal impedance of the power supply, and R4 the load.R2; R3; C1; C2 are parasitic factors of transmitting and re-ceiving coils. This circuit is mathematically modelled as thefollowing state equation. This type of expression is widelyused in control theory. In particular one can write the model ofWPT circuits in a compact form, and obtain a clear perspectiveto analysis of stability and responses with a sinusoidal input.

_x = Ax+Bu

x =

2664v1v2i1i2

3775

A =1

26640 0 C1 0

0 0 0 C2L2 M2 (R1 +R2)L2 (R3 +R4)M2M1 L1 (R1 +R2)M1 (R3 +R4)L1

3775

B =1

266400L2M1

3775 = L1L2 M1M2: (3)

C. Formulation of average power and efficiencyThe purpose of this paper is to investigate the relation

between the frequency of AC supply voltage, the averagepower delivered to the receiving side, and the efficiency ofaverage power transmission. The efficiency is defined as theratio of the average power obtained at the receiving sideagainst the average power supplied at AC power supply. Thus

we formulate equations of the average powers at transmittingand receiving sides, and of the efficiency in the following.

Since the matrix A in the equation (3) is stable, i.e., alleigenvalues of the matrix A have negative real parts, thesolution to the equation (3) will be stationary after time havepassed adequately. The stationary solution with a sinusoidalinput u = sin!t, will have the same frequency of the input.

In general, the average powers P1 and P4 which are atthe AC supply and the load in the receiving side respectively,and then the efficiency depends on the frequency of ACsupply. These stationary values can be all expressed in termsof transfer functions.

P1 =1

2

Re[G1(j!)]R1jG1(j!)j2P4 =

1

2R4 jG2(j!)j2

=P4P1

: (4)

Here G1(s) and G2(s) are the transfer functions from theinput u to i1 and i2, respectively. With the equation (3),these transfer functions are written in G1(s) = H1(sI A)1B;G2(s) = H2(sI A)1B; where H1 = [0; 0; 1; 0]and H2 = [0; 0; 0; 1].

D. Resonance and average power

In view of the equation (4), we see that if one puts an ACinput voltage with the frequency which gives the maximal gainof the transfer function G2(s) (the gain is called H1-norm),one will have the maximal average power at the receiving side.To illustrate the situation, we set values of circuit elementsas below. These values are given by consulting a practicalsituation of WPT[2].

TABLE IPARAMETERS

elements values elements valuesR1 50 L2 10HR2 0:1 M1 0:5HR3 0:1 M2 0:5HR4 50 C1 1nFL1 10H C2 1nF

The bode diagram of G(s) is shown as below.

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Figure 2 Bode Diagram

In this case the resonance will occur at the unique frequency! = 1:00 107rad=sec and we have the maximal averagepower at the receiving side if we use a sinusoidal wave withthe resonant frequency.

E. Resonance and efficiencyBased on the equation (4), we can observe that the relation

between the input frequency and the efficiency of WPT isnot straightforward. Many other papers state that using anAC input voltage with a resonant frequency is optimal orbetter in the design of WPT. In the previous section, we haveclarified that using the input with resonance maximizes theaverage power at the receiving side. However, the efficiencyof power transmission can be maximized either when oneuse resonance or when one use nonresonance. This difficultsituation is illustrated by numerical examples in the following. First, power P4 and efficiency with the condition ofTABLE I are shown as below.

Figure 3 Power and Efficiency for TABLE I

Then, is maximized when

! = 1:07 107rad=sec (5)as previously known in [9].

Another example is shown as below. The values of elementsare set as TABLE II. The only difference between TABLE Iand TABLE II is the value of C1.

TABLE IIPARAMETERS

elements values elements valuesR1 50 L2 10HR2 0:1 M1 0:5HR3 0:1 M2 0:5HR4 50 C1 0:1nFL1 10H C2 1nF

Figure 4 Power and Efficiency for TABLE II

By comparing Figure 3 and Figure 4, we see that thefrequencies which respectively maximize power and efficiencyare different.

III. CONCLUSION

In this paper, we have pointed out that using a resonantfrequency of AC voltage input does not lead to maximize theefficiency of average power transmission, although resonanceis equivalent to maximization of output average power, in gen-eral WPT systems. In fact, we have illustrated a situation thatnon-resonance maximizes the efficiency of power transmissionby numerical examples. This suggests that we should take bothof output power and efficiency into account and then decidea balanced working frequency, in the case that resonance isnot equivalent to the maximal efficiency. For example, themaximum power is desirable if an AC power supply can serveenough power, but the maximum efficiency is best if the supplycan serve less power.

Even one of the simplest WPT circuits treated in this papermay have disagreement between resonance and efficiency.Therefore, one will face on further difficulty to design betterWPT when one tries more complex circuit with more elementsin order to meet an increasing design specification.

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REFERENCES[1] Andr Kurs, Aristeidis Karalis, Robert Moffatt, J. D. Joannopoulos, Peter

Fisher, Marin Soljacic , Wireless Power Transfer via Strongly CoupledMagnetic Resonances, Science 317, PP.83-86, 2007.

[2] Huy Hoang, Franklin Bien, Maximizing Efficiency of ElectromagneticResonance Wireless Power Transmission Systems with Adaptive Cir-cuits, Wireless Power Transfer - Principles and Engineering Explo-rations, PP.207-226, 2012.

[3] Ick-Jae Yoon, Hao Ling, Realizing Efficient Wireless Power Transferin the Near-Field Region Using Electrically Small Antennas, WirelessPower Transfer - Principles and Engineering Explorations, pp.151-172,2012.

[4] Ji-Yeon Mun, Min-Gyeong Seo, Woo-Geun Kang, Hae-Young Jun, Yong-Ho Park, Jeong-Ki Pack, Study on the Human Effect of a Wireless PowerTransfer Device at Low Frequency, PIERS Proceedings, pp.322-324,2012.

[5] Hiroshi Hirayama, Equivalent Circuit and Calculation of Its Parame-ters of Magnetic-Coupled-Resonant Wireless Power Transfer, WirelessPower Transfer - Principles and Engineering Explorations, pp.117-132,2012.

[6] Youngjin Park, Jinwook Kim, Kwan-Ho Kim, Magnetically CoupledResonance Wireless Power Transfer (MR-WPT) with Multiple Self-Resonators, Wireless Power Transfer - Principles and Engineering Ex-plorations, pp.51-64, 2012.

[7] Henri Bondar, Shailendra Oree, Zafrullah Jagoo, Keiichi Ichikawa, Esti-mate of the maximum range achievable by non-radiating wireless powertransfer or near-eld communication systems, Journal of Electrostatics,Volume71, Issue4, pp.648-655, 2013.

[8] Marco Dionigi, Alessandra Costanzo, Mauro Mongiardo, Network Meth-ods for Analysis and Design of Resonant Wireless Power TransferSystems, Wireless Power Transfer - Principles and Engineering Explo-rations, pp.65-94, 2012.

[9] Takashi Komaru, Masayoshi Koizumi, Kimiya Komurasaki, TakayukiShibata, Kazuhiko Kano, Compact and Tunable Transmitter and Receiverfor Magnetic Resonance Power Transmission to Mobile Objects, Wire-less Power Transfer - Principles and Engineering Explorations, pp.133-150, 2012.

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