On Wireless Power Transfer

Lucia Dumitriu1, Dragos Niculae1, Mihai Iordache1, Lucian Mandache2 and Georgiana Zainea1

1 Politehnica University of Bucharest, Romania, e-mail dlucia@elth.pub.ro 2 University of Craiova

Abstract— The paper is a general survey on power transfer by induction. The physical principle is pointed out and the constraints for an optimal operation applied both to the emitter and receiver are specified. Couple mode theory is used to study the wireless power transfer and a comparison with the circuit theory results is made. The analysis is fo-cused on the power transfer efficiency in connection with the mutual inductance. An accurate method for mutual in-ductance computation is presented and the power transfer optimization is performed for different geometrical configu-rations which involve different mutual inductances.

I. INTRODUCTION Although the applications in telecommunication area

are based on the propagation of electromagnetic waves, the antenna radiation technology is not suitable for power transfer. The main reason is that the radiated elec-tromagnetic power is small (a vast majority of the energy is wasted by dispersion into the free space) making this technology more suitable to transfer information rather than power. In electromagnetic field theory two concepts appear: the near field,considered as a non-radiative type that occurs close to the antenna at a distance smaller than one wavelength and decays very fast (~1/r3) and the far field, considered of a radiative type. The last one propa-gates starting from a distance equal to two wavelengths from antenna up to infinity. This type of radiation decays much slower than the near field (~1/r). The emitted power decays with the square of the distance [1].

There is a transition zone starting from a distance of one wavelength from the antenna up to two wavelengths in which the combined effects of the near and far fields occur.

Wireless power/energy transfer or shortly Witricity (WIreless elecTRICITY) is different from wireless trans-mission of information. In this new technology (useful in cases where instantaneous or continuous energy is needed but interconnecting wires are inconvenient, hazardous, or impossible), the transfer is made over distances at which the electromagnetic field is strong enough to allow a rea-sonable power transfer. This is possible if both the emit-ter and the receiver achieve magnetic resonance.

The physical principle the Witricity concept is based on is the near-field in correlation to the resonant induc-tive coupling. This implies that the coupled systems work at their resonance frequency. The reason is that the reso-nant objects exchange energy efficiently while the non-resonant objects interact weakly.

The Witricity system consists of two resonators – source and device (load)– which theoretically could be in

one of the following connections: series-series, parallel-parallel, series-parallel, and parallel-series.

The source resonator emits a lossless non-radiative magnetic field oscillating at MHz frequencies, which mediates an efficient power transfer between the source and the load resonator.

Some remarks have to be made [2]: - The interaction between source and device is strong

enough so that the interactions with no resonant objects can be neglected, and an efficient wireless channel for power transmission is created.

- Magnetic resonance is particularly suitable for appli-cations because, in general, the common materials do not interact with magnetic fields.

- It seems [3] that the power transfer is not visibly af-fected when humans and various objects, such as metal, wood, electronic devices, are placed between the two coils at more than a few centimetres from each of them, even in cases where they completely obstruct the line of sight between source and device.

- Some materials (such as aluminium foil and humans) just shift the resonant frequency, which can in principle be easily corrected with a feedback circuit.

In the last decade, advances in wireless communication and semiconductor technology have produced a wide variety of portable electronic, medical and industrial de-vices. However, the mobility degree of these devices is strong relied on how often you have to manually plug in for recharging their batteries. Furthermore, as portable devices shrink, connectors become a larger fraction of system size. Thus, a growing interest of researchers, fo-cused on implementation of wireless technologies in bat-teries recharging, was emerged. Wireless power offers the possibility of connector-free electronic devices, which could improve both size and reliability.

Radiative transfer, although perfectly suitable for transferring information, poses a number of difficulties for power transfer applications: the efficiency of power transfer is very low if the radiation is omni-directional, and unidirectional radiation requires an uninterrupted line of sight and sophisticated tracking mechanisms [4].

In some recent papers, various researchers have been trying to transfer energy using wireless technologies like: laser beam, piezoelectric principle, radio waves and mi-crowaves, inductive coupling [5].

The principle behind the antenna radiation technology was described by the Coupled-mode Theory, which re-cently was extended to describe wireless power transfer

978-1-4673-1810-5/12/$31.00 ©2012 IEEE

[6-15], but, because RLC resonators are used, the Circuit Theory can be applied as well.

The paper presents the two approaches for wireless power transfer analysis: the Couple-mode Theory and the Circuit Theory in a comparative study. The analysis is focused on the power transfer efficiency in connection with the mutual inductance. An accurate method for mu-tual inductance computation is presented and the power transfer optimization is performed for different geometri-cal configurations which involve different mutual induct-ances.

II. SERIES-SERIES RESONATORS INDUCTIVELY COUPLED

A. Analysis by Circuit Theory Let us consider two series-series resonators inductively

coupled as in Figure 1, where L3 and L4 are two coaxial identical coils represented in Figure 2, [13, 14]. The pa-rameters of the coils are: the radius r = 150 mm, the pitch p = 3 mm, the wire size w = 2 mm, the distance between the coils g = 150 mm, and the number of the turns N = 5. Using the Q3D Extractor program [15], we get the follow-ing numerical values for the parameters of the system of the two inductively coupled coils:

nF0404.121 === CCC , H747.163 µ=L ,

H736.164 µ=L , M = 1.4898 µH, R5 = 0.12891 Ω, R6 = 0.12896 Ω. The source circuit has the resistance R7 = 5.0 Ω, and the

load is R8 = 5.0 Ω.

Figure 1. System of two series-series resonators driven by a voltage

source

p

r

Transmitter

g

Receiver

Figure 2. Coil geometry

We build the state equations of the circuit in normal-symbolic form:

11

1

1d

dL

C iCt

v= (1)

2

2

2

1d

dL

C iCt

v=

(2)

( )

( )92

43

42

43

86

243

7542

432

43

4

2

1211

d

d

eMLL

Li

MLL

RRM

iMLL

RRLv

MLLMv

MLL

Lt

i

L

LCCL

−+

−

++

+−

+−

−+

−−=

(3)

( )

( )92

432

43

863

243

752

43

32

43

2

1212

d

d

eMLL

MiMLL

RRL

iMLL

RRMv

MLL

Lv

MLLM

t

i

L

LCCL

−−

−

+−

−−

++

−−

−=

(4)

Considering null initial conditions, i.e.:

( ) ( ) ( ) ( ) A00 andA00V00V002121

, , ==== LLCC iivv (5)

and taking the input voltage ( ) ( )Vcos100 09 tte ω⋅= , with

( )( ) CLL ⋅+=ω

2/1

430 , if we integrate the state equa-

tions for the numerical values of the circuit parameters given above, we get the time-domain response. As exam-ple, the capacitor voltages are shown in Figure 3.

1Cv

2Cv

Figure 3. Time variation of the voltages vC1 and vC2 obtained by

integrating the state equation

B. Analysis by Couple-mode Theory In order to obtain the differential equations of the cou-

pled-mode amplitudes, the equations (2) and (4) are mul-tiplied by 2/C , and the equations (3) and (5) by

2/3Lj and 2/4Lj , respectively. Adding the first and the third equations so modified, we get:

912121110

91''

212'11

"11

1

)()()(

)()(d

)(d

ektaktataj

ektakatajtta

e

e

++γ−ω−≅

≅++γ−ω−= (6)

where:

( ) ( )

( )2

43

7541

31

243

4312

43

312

24

486

''2

13'

1

13

243

43''1

4d

23d

1

;1

2;

22

2

22

22:

22

22

1

11

2211

MLL

RRL

CL

MLL

LLjk

MLLM

CL

jk

aiL

LCRRjjvCa

aiL

ja

aiL

jvCMLL

LLa

iL

jvCaiL

jvCa

e

LC

L

LC

LCLC

−

+=γ=ω

−=

−=

≅+−+=

≅=

≅+−

=

+=+=

(7)

Adding the modified second equation and the modified fourth one, we get:

921212220

92'121

'22

"22

2

)()()(

)()(d

)(d

ektaktataj

ektakatajt

ta

e

e

++γ−ω−≅

≅++γ−ω−= (8)

where:

( )

243

632

42

243

422

43

421

13

35

'1

24'

2

24

243

43''2

;1

2;

22

2

22

11

2

22

MLL

RL

CL

MLLML

jkMLL

MCL

jk

aiL

LCjRjvCa

aiL

ja

aiL

jvCMLL

LLa

e

LC

L

LC

−=γ=ω

−−=

−=

≅−+=

≅=

≅+−

=

(9)

The solutions of the differential equation system built with eq. (6) and (8) can be obtained by integration and they have the general form:

tjtata 0ee:)( 0ωγ−= (10)

For small loss, because 2a (which is the circuit en-ergy) decays as ( )tγ− 2exp , we can express the dissipated power as a function of the square of the mode amplitude:

Pd _ cmt = −dW (t)

dt= 2γW (t) = f (a(t)

2) (11)

C. Transfer Power Computation We can compute the power dissipated on the load resis-

tance as Joule power

( )2__8__ 28 sssvLsssvR iRp = (12)

If we use the mode amplitude, we can express this power either by the equation

)2abs( *__2__22___8 sscmtsscmtssaproxcmtR aap ⋅⋅γ⋅−= (13)

or by

( ) ( )( )tatat

p ssdcmtR*22___ d

d8

⋅= (14)

We can also compute the current through the load circuit

ss_cmt_Li 2 from the expression of ssa _2 by identifying

its imaginary part. With this value we compute the Joule power dissipated on load:

( )2__8__ 28 sscmtLsscmtR iRp = (15)

The graphical representation of these instantaneous pow-ers is given in Figure 4.

sscmtRp __8

ssaproxcmtRp ___8 ssdcmtRp ___8

sssvRp __8[W]

Figure 4. Time variation of the transferred power in the four approaches

We remark a significant difference between the power computed in Circuit Theory (CT) and those obtained by the Couple-Mode Theory (CMT) formulas. This is a con-sequence of the approximations made in the expressions of a1 and a2.

The time-averaged transferred powers on a period 02 /2 ωπ=T , with n =100000, have the following values:

n = 100000; T2 = 8.28814.10-7 s; PR8_med_sv_ss = 134.78914 W;

PR8_med_cmt_ss = 34.2224 W;

PR8_med_cmt_aprox_ss = 69.7736 W;

PR8_med_cmt_d_ss = 15.796511 W

(16)

Remark: As expected, the difference between the in-stantaneous powers (Figure 7) is projected over the time-averaged powers.

By using CT and either Laplace transform or complex representation, and symbolic analysis, we can obtain in-formation regarding the variation of the power transferred on the circuit load with respect to different parameters needed in the optimized design. Keeping as variables only the frequency and the mutual inductance, we can study the influence of these parameters on the transferred power value; for instance, in Figure 5 is shown the dependence of the transferred power with respect to the frequency.

Figure 5. Power variation with respect to frequency.

We can see that the power has two maxima: 7066.472_1max_8

=ssfRP W and 7677.476_2max_8

=ssfRP W at the frequencies 6

_1max_ 1016183.18

⋅=ssPRf Hz and 6

_2max_ 1025798.18

⋅=ssPRf Hz, respectively, and a minimum 9381.268min__8

=ssRP W at the frequency 6

min__ 1020798.18

⋅=ssPRf Hz. This situation is called frequency splitting.

The quality factor of the circuit is

643.81014.11028.1

1021.166

6

__

_01_ =

⋅−⋅⋅

=−

=ssisss

ssssc ff

fQ (17)

where f0_ss = 1.21.106 Hz, while fi_ss, and fs_ss are identi-fied by the intersection of the curve PR8_f_ss with

( ) 0.2/max __8 ssfRPy = , and have the values fi_ss = 1.14 MHz and fs_ss = 1.28 MHz.

III. MUTUAL INDUCTANCE COMPUTATION In order to compute the mutual inductance for different

situations occurring in practice, many formulas have been developed. The most known for coaxial windings was developed by Maxwell [16]. Other approaches belong to Weinstein [18] (an expression with power series of the complementary modulus), Nagaoka [17] (he proposes formulas based on Jacobi’s q-series [19]), Havelock (for-mulas based on certain definite integrals of Bessel func-tions), Mathy (formulas with elliptic integrals of the third kind), Coffin, Rowland, and Rayleigh [18].

To perform the analytical calculation of the mutual in-ductances between different magnetically-coupled coils, we developed a Matlab tool based on the Neumann for-mula [20]. It uses two grids for each winding – one inside and another outside (Figure 6). In this manner we can limit the mutual inductance value between a maximum and a minimum. The final value of the mutual inductance will be the average of two values, corresponding to the two grids. For a better comparison, we have chosen from the literature the four cases described in Figure 7.

We denote by Mp the mutual inductance value com-puted by the above procedure. We have also simulated the four configurations using Ansoft Q3D Extractor program. The simulation results are denoted by MQ3D.

Figure 6. Description of the grids corresponding to the coils [20].

Case 1 Case 2 Case 3 Case 4

Figure 7. Geometry of the four cases

The simulation results were compared with experimen-tal ones. Using two handmade coils, we have got by measurement the mutual inductance value Mm in the four configurations. The coils have 16 turns of insulated solid copper of rectangular cross-section 3.2mm x 1.8mm, with the coil radius of 100mm (Fig. 8).

The results of Mp, MQ3D and Mm are given in Table I for different relative positions between coils.

Figure 8. Experimental model.

TABLE I. COMPUTED AND MEASURED MUTUAL INDUCTANCES

Case 1

mm30=x

Case 1

mm100=x

Case 3

30105arctg

mm30

=θ

=x

Case 4

o45

mm30

=θ

=x

Mm 51,97 µH 16,19 µH 17,61 µH 14,73 µH Mp 50,33 µH 15,92 µH 17,01 µH 16,52 µH

MQ3D 47.483 µH 14.823 µH 15.734 µH 16.799 µH

One must notice that the simulations and the tests have been performed at 50Hz. We repeated the Ansoft Q3D Extractor simulations at 10MHz, and the values of the self inductances, the capacitances, and the mutual inductances

remained unchanged, but the resistance values were big-ger, because of the skin effect.

IV. OPTIMIZATION OF THE ACTIVE POWER TRANSFFER To optimize the wireless transfer of energy based on the

configurations above, we have to deal with some restric-tions:

1) For a given configuration, with fixed geometrical pa-rameters, the maximum transfer of energy is provided when the operation frequency is near the self-resonance frequency.

2) Keeping a constant value of the magnetic coupling in device, while the distance x between the two coils varies, using a Matlab procedure we obtain the radius values of the receiving coil R2 given in Table II:

TABLE II. DISTANCE BETWEEN COILS AND RECEIVING COIL RADIUS VALUES FOR

FIXED TRANSMITTER COIL R1=0.150M AND FIXED MUTUAL INDUCTANCE M=65.2µH

x [m] 0 0.01 0.03 0.05 0.07 0.09

R2 [m] 0.100 0.100

0.300 0.110 0.290

0.120 0.280

0.140 0.260 0.200

This is an evident result, knowing that the mutual in-

ductance can be expressed as:

2

121

1

212 ,i

NM

iN

Mϕ

=ϕ

= (18)

The magnetic flux is a surface integral of magnetic field vector and to maintain it at a constant value, the surface (the radius also) follows the field line (see Figure 9).

Figure 9. The geometrical evolution of receiving coil for a constant

magnetic coupling

3) If the main goal is to obtain the maximum magnetic coupling for different values of the distance between the coils, the optimum values of the radius of receiving coil are plotted in Table III for R1=0.150m and M=max. In this case, for a fixed value of the distance between the two coils, for a maximum coupling, the receiver coil grows trying to enclose as many as possible magnetic field lines (see Figure 10).

TABLE III. OPTIMUM VALUES OF THE RADIUS OF RECEIVING COIL FOR FIXED R1 AND

MAXIMUM MUTUAL INDUCTANCE

x [m] 0.01 0.03 0.05 0.07 0.09 R2 [m] 0.150 0.160 0.170 0.190 0.200 M [H] 2.14e-4 1.32e-4 9.85e-5 7.84e-5 6.5e-5

Figure 10. The geometrical evolution of the receiving coil for a

maximum magnetic coupling

4) If the second coil is translated along a plan at a given distance x, the maximum of the mutual inductance is ob-tained for coaxial position of the two coils (see Figure 11).

Figure 11. The variation of the mutual inductance with the displacement

of the receiving coil.

5) If the angle between the two coils is modified, the shape of the mutual inductance variation depends on the distance x (see Figure 12).

mx 6.0=

mx 0.1=

Figure 12. The variation of the mutual inductance with angle and the

distance between the two coils.

V. CONCLUSIONS Concluding the study we can make the following re-

marks: 1. Wireless power transfer is a new technology, useful

when electrical energy is needed but interconnecting wires are, for certain reasons, impractical or impossible; the

transfer can be made over distances at which the electro-magnetic field is strong enough to allow a reasonable power transfer. This is possible if both the emitter and the receiver achieve magnetic resonance, because the resonant objects exchange energy more efficiently than the non-resonant ones.

2. The modern applications in telecommunications (in-formation transfer) are based on the propagation of elec-tromagnetic waves, but the antenna radiation technology is not suitable for power transfer because of the weak effi-ciency (a vast majority of the energy is wasted by disper-sion into free space).

3. The principle behind the information transfer can be described either by Maxwell’s equations or by the Cou-pled-mode Theory (CMT). In wireless power transfer, because RLC resonators are used, the Circuit Theory can be successfully applied.

4. The main advantage of CMT is reducing the number of differential equations which describe the circuit behav-ior to half. In the case of the cascade resonators this in-volves a significant decrease in the computational effort.

5. The assumptions made in CMT regarding • the frequency interval on which the study is per-

formed, which has to be small enough so that the coefficients which arise in the mode amplitude equations are constant (independent of frequency)

• the coupling factor k, which needs to be small enough so that the resonance frequency splitting doesn’t occur, have to be considered when one in-tends to apply this analysis.

6. The power transferred on the load resistance can’t be accurately computed by means of the mode amplitudes.

7. CT offers an important analysis tool for linear or nonlinear circuits which is the state variable approach [21]. The analysis performed by this method gives exact information about the power transfer, which is the great advantage of the Circuit Theory by comparison with Couple-mode Theory.

8. Using a suitable symbolic simulator [16], we can get useful information for optimizing the design of the wire-less power transfer system.

9. The efficiency of the wireless transfer power deeply depends on the resonator parameters (self inductances, mutual inductances, capacitances, and ohmic resistances of the two magnetically-coupled coils). Consequently, parameter identification is a very important objective in the automatic design of such a system. In this chapter an efficient procedure to compute the mutual inductance is developed and the results of its implementation in Matlab are compared with those obtained by using the program Ansoft Q3D Extractor. We note a good agreement.

10. The measurements made using a device with two copper coils in various positions give similar results. The differences between the measured values and the ones computed could be explained by apparatus accuracy and by the fact that the integration procedure neglects the re-partition of coil turns, considering a single medium turn.

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