a 09 - pag 45-50 - niculae - wireless

ISSN 1843-6188 Scientific Bulletin of the Electrical Engineering Faculty Year 11 No. 2 (16)

45

WIRELESS POWER OPTIMAL TRANSFER IN MAGNETIC

COUPLED RESONATORS

D. NICULAE1, L. DUMITRIU1, M. IORDACHE1, L. MANDACHE2, G. ZAINEA1 1 Politehnica University of Bucharest, Spl. Independentei 313, 060042, Bucharest, Romania,

2 University of Craiova, E-mail: [email protected]

Abstract. Witricity (WIreless elecTRICITY) represents an experimental technology used to transfer electricity/power between electrical sources and receivers without using wires. The transfer is made over distances at which the electromagnetic field is strong enough to allow a reasonable power transfer. This is possible if both the emitter and receiver

achieve magnetic resonance. Our paper proposes an efficient method to determine the equivalent voltage generator in respect of the output port of the two magnetic coupled resonators. In this way we can compute very easy the resonant frequency of the resonators and the maximum active power transferred from

the emitter resonator to the receiver resonator.

Keywords: Near field, wireless transfer power, magnetic coupled

resonators, equivalent voltage generator, and maximum active

power.

1. INTRODUCTION

The electric signals are basis concepts of the electromagnetic field. They are carriers of energy and information and the applications take this into account. Concepts like near and far fields, and radiative propagation related to the antenna concept occur when study the physical principle these applications are based on. The near field is referred to as a non-radiative type that occurs close to the antenna in a distance smaller than one wavelength and decays very fast (1/r3). The far field is considered to be of a radiative type. It propagates starting from a distance equal with 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 distance square. 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, [1-22]. The modern applications in telecommunication area are based on propagation of electromagnetic waves (on far field), but the antenna radiation technology is not suitable for power transfer. The main reason is that the radiated electromagnetic 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 than power. Witricity (WIreless elecTRICITY) represents an experimental technology used to transfer electricity/power between electrical sources and receivers without using wires. The transfer is made over distances at which the electromagnetic field is strong

enough to allow a reasonable power transfer. This is possible if both the emitter and receiver achieve magnetic resonance. Wireless transmission is useful in cases where instantaneous or continuous energy is needed but interconnecting wires are inconvenient, hazardous, or impossible. The term Witricity was introduced by Professor Marin Soljai from MIT who, together with his team, started to work on the subject in 2005. Their first papers on this subject, "Coupled-mode theory for general free-space resonant scattering of waves" [2], and "Wireless power transfer via strongly coupled magnetic resonances" [3] was published in 2007. An experimental demonstration is presented, in which a 60 W light bulb was powered wirelessly over a distance of 2 m with an efficiency of 45%. The coils resonated together at 9.9 MHz. Many of the Witricity technology aspects can be found in Andre Kurs Master of Science Thesis [4], or Bachelor of Science Thesis of Robert Moffatt [5] members of the initial team who developed the technology. In order to better promote the Witricity concept, a company was founded shortly after the paper publication by Soljai and some members of his team [6]. At the beginning the technology was proposed for small power application like cordless battery charging of laptops and mobile phones. The papers that succeeded, suggested practical implementations of the Witricity concept in electric vehicle field [7] and medical area [8 , 9] as a way to recharge the otherwise unreachable sensors and implanted devices. The coupling through magnetic resonance implies the coupled systems to work at their resonance frequency. In [2], the energy is transferred between two simple coils which form resonators due to the parasitic capacitance of their turns. In [9] the concept is improved the coils are connected to capacitors leading to retention of the electric field inside those devices. The coupling is made through magnetic field and the electric field is reduced. Some remarks have to be made [9-21]: The interaction between the source and device is

strong enough so that the interactions with non-resonant objects can be neglected, and an efficient wireless channel for power transmission is built;

Magnetic resonance is particularly suitable for applications because, in general, the common materials do not interact with magnetic fields;

It seems [2] that the power transfer is not visibly affected when humans and various objects, such as

Scientific Bulletin of the Electrical Engineering Faculty Year 11 No. 2 (16) ISSN 1843-6188

46

metals, wood, electronic devices, are placed between the two coils at more then 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.

Obviously, the efficiency of the wireless transfer power depends of the configuration of the two magnetic coupled resonators (series-series, parallel-parallel, series-parallel and parallel-series), of the values of the two circuit parameters, and of the resonant frequency. It was proved that the best configuration is the serier-series configuration. In this paper we propose an efficient method to determine the equivalent voltage generator in respect of the output port of the two magnetic coupled resonators. In this way we can compute very easy the resonant frequency of the resonators and the maximum active power transferred from the emitter resonator to the receiver resonator.

2. WITRICITY POWER TRANSFER IN A SERIES-

SERIES CONFIGURATION

Let us consider two series-series resonators inductively coupled as in Fig. X-1, in which the independent current sources j2 is not connected. The coaxial identical coils L3 and L4 are represented in Fig. xx. The parameters 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 program Q3D Extractor we have obtained the following numeric value for the two coils:

0404.121 === CCC nF, 747.163 =L H, 736.164 =L

H, M = 1.4898 H, R5 = 0.12891 , R6 = 0.12896 , R7 = 5.0 , and R8 = 5.0 .

Figure 1. Coupling of two series-series resonators driven by

a voltage source.

Figure 2. Parameters of coils [7].

For any configuration resonator, studied in chapter four, we can compute the maximum active power transfer from the first circuit to the second one. We consider only the configuration series-series represented in Figure 1. For the witricity power transfer system in Figure 1 the condition 0== el XX implies

2 2: (1 ( 1 2 32 2 6 2 22 4 1 3 ) ( 1 3

22 1 2 4 5 72 22 2 4 1 3 1 2 4 7

2 2 2 41 2 1 2 4 5 )2 2 2 2( 5 1 7 1 2 4 2 1 3

2 2 4 2 22 5 1 7) ) / ( 2(1 1 3

( 2 1 3

Xl M C C L

C L C L C L

C C L R R

C L C L C C L R

M C C C C L R

R C R C C L C L

R C R C C L

C L

= + = + = + = +

+ + + +

+ + + +

+ + + +

+ + + +

+ + ++ + ++ + ++ + +

+ + ++ + ++ + ++ + +

+ + + + 2 2 25 1 2 5 1 72 2 27 1 ) ))

R C R C R

R C

+ + + + + + + +

(1)

in which is the frequency of the voltage source. If the circuit parameters are given, we can compute the resonant frequencies. Substituting the load resistance Rl = R8 by an ideal independent current source J2 we can compute, using the SYMNAP (SYmbolic Modified Nodal Analysis Program), [16, 22], program, the equivalent voltage generator in respect of the terminals o o in full symbolic form. Running the SYMNAP program we get:

20_ _ _ _ : 24649.000

1 9 / 3925. 5 1 625.

224649. 1 3 3925. 7 1

VAB sim f th ss M f

C e I R f C

f C L I R f C

= = = =

+ + + +

(2)

and

p

r

Transmittin

g coil

g

Receiving

coil


47

70 010000000 10

10 12099522294 10 039250000 10

2 111 3 062500000 10 7 1 024649000

13 3 12 210 2 4 5 1 039250000 10 2 4

14 4015479572 10 2 4 1 3

ZAB _sim_f_Th_ss : .

( . I . I

f C L . R f C .

f C L R C . I f C L

. I f C L C L

= = = =

+ + + +

+ + + +

+ + + +

13 3024649000 10 2 4 7 1

12 2039250000 10 2 6 5 1

11062500000 10 2 6

13 3024649000 10 2 6 1 3

12 2039250000 10 2 6 7 1

14 2 4015479572 10 1 2

062500000

. f C L R C

. I f C R R C

. f C R

. f C R C L

. I f C R R C

. I M f C C

.

+ + + +

+ ++ ++ ++ +

+ + + +

+ + + +

+ ++ ++ ++ +

+ ++ ++ ++ +

+ + + + 1110 5 1 3925 5 1 625

224649 1 3 3925 5 1

R f C )/ ( . I R f C .

. f C L . I R f C ))

+ + + +

+ + + +

(3)

Remark:

We can see that all the parameters of the Thvenin generator are functions of all the circuit parameters (L3, L4, C1, C2, M, R5, R6, R7, and R8) and also of the frequency. The numeric value of the input signal is E8 = 100j, and the resonant frequency

is ( )( ) 2065435.12/2/1 430 =+= CLLf MHz. Separating the real and the imaginary part of the equivalent impedance we get:

))171015405625.0

7151030811250.0

151015405625.0

311060757320.0

311030811250.0.390625(2/(

/)5211012151464.0

7211012151464.01

7621030811250.015

621030811250.0715

621061622500.051.79

31621012151464.0

316210616225.0

621078125.071.79(

000000500000.0:____0

2228

228

2228

2249

28

24214

242142

221222

2122

212

2414

212

10

CfR

RCfR

CfR

LCf

LCfC

RCCfM

RCCfMC

RRCfCR

RCfRCR

RCfRC

LCRCf

LCRCf

RCRC

ssThfsimRAB

++++

++++++++

++++++++

++++++++

++++

++++

++++++++

++++

++++

++++

++++

++++

++++

====

(4)

and

50 0 20000000 10

11 13 20 31100717 10 0 24531250 10 1 3

14 4 2 20 483736621 10 1 3 0 12265625

13 2 14 410 2 4 0 96747325 10 2 4

16 6 2 21 3 0 19077799 10 2 4 1 3

0 48

XAB _sim_f_Th_ss : .

( . . f C L

. f C L .

f C L . f C L

C L . f C L C L

.

= = = =

+ + + +

+ + + +

+ + + +

14 2 4373662 10 1 2

16 2 6 20 19077799 10 1 2 3

13 2 2 20 12265625 10 7 1

13 2 20 24531250 10 5 1 7

14 4 20 96747325 10 1 2 4 5 7

14 4 2 20 48373662 10 1 2 4 5

10 48373662 10

M f C C

. M f C C L

. R f C

. R f C R

. f C C L R R

. f C C L R

.

+ + + +

+ ++ ++ ++ +

+ ++ ++ ++ +

+ + + +

4 4 2 21 2 4 7

13 2 2 20 12265625 10 5 1

8 22 390625 0 30811250 10 1 3

9 4 2 2 80 60757320 10 1 3 0 15405625 10

2 2 2 8 2 25 1 0 30811250 10 5 1 7

8 2 2 20 15405625 10 5 1

f C C L R

. R f C ) /

/(f C ( . . f C L

. f C L .

R f C . R f C R

. R f C ))

+ + + +

+ + + +

+ + + +

+ + + + + + + +

+ + + + + + + +

+ + + +

(5)

Computing the Thvenin generator parameters at the resonance frequency 0ff = we obtain:

)427324)43(

)43(3423

2:___0_0

7

5

5/(9)43(2

ILCRILCLCLL

CLLLILCILC

ssthffVAB

R

R

ReMCLL

+++

+++

=

+

+

+

(6)

and

))43)(4273

274)43()43(3

425325(2(

/)427327425

3254)43(76

23)43(7624

)43(5623)43(

5623)43(

4)43()43(

342)43(4(

2)43(:__0__0

222

2

2

2

2

CLLILCRILC

RLCLLCLLL

ILCRILCR

LCRLCRLCR

LCRLCLLRR

CILCLLRRCIL

CLLRRCILCLL

RRCIILCLL

ILCLLCLL

LLICLLMI

CLLssThfsimZAB

++++++++

++++++++++++++++

++++

++++

++++++++

++++++++++++

++++++++++++

++++++++++++

++++++++++++++++

++++

++++====

(7)

Consequently


48

))43)(427327

4)43()43(34

25325(2/()427

327425325

4)43(7623)43(

7624)43(562

3)43(562

3)43(4)43(

)43(342)43(

4(2)43(:__0_0

2

222

22

2

CLLILCRILCR

LCLLCLLLIL

CRILCRLCR

LCRLCRLCR

LCLLRRCILCLL

RRCILCLLRRCI

LCLLRRCI

ILCLLILCLL

CLLLLICLL

MICLLssThfRAB

++++++++++++

++++++++++++++++

++++

++++++++

++++++++++++++++

++++++++++++

++++++++++++

++++++++++++++++++++

++++++++++++

++++====

(8)

and

0:_)0_0 =ssThfXAB (9) The symbolic expression of the maximum power transferred to the load resistance is

28_max_ 0_ _ : 1.0000000 ( 3 4) /

( 5 2 3 5 2 4

3 ( 3 4) ( 3 4) 4

7 2 3 7 2 4 ) ( 3 4)

2(4 ( 3 4) 2 4 3 ( 3 4)

2 2( 3 4) 4 ( 3 4) 3

2

PR f Th ss L L

R C L I R C L I

L L L C L L C L

R C L I R C L I L L C

I M L L C I L L L L C

L L C L I L L C L I

I C

= += += += +

+ + + +

+ + + + + + + + + + + + + + + +

+ + + + + + + + + + + +

+ + + + + + + + + + + +

+ + + ++ + + ++ + + ++ + + +

+ + + + 6 5 ( 3 4) 3

2 6 5 ( 3 4) 4 2 6 7

( 3 4) 3 2 6 7 ( 3 4) 4

2 2 25 2 3 5 2 4 7 2 3

2 2 27 2 4 2 6 3 2 6 4 ))

R R L L C L

I C R R L L C L I C R R

L L C L I C R R L L C L

R C L R C L R C L

R C L C R L C R L

+ + + + + + + +

+ + + + + + + + + + + +

+ + + + + + + + + + + + + + + +

+ + + + + + + +

+ + + +

(10)

The numeric values of these parameters at the angular

frequency 60 105771.7 = [rad/s] are:

[ ]

[ ]

[ ]=

=

=

.0:_)0_0

973397.24:__0_0

08540.220:__0_0

ssThfXAB

ssThfRAB

VssThnfVAB

(11)

and the maximum power transferred to the load resistance is:

[ ]WsscThfPR 89182.484:__0max__8 = (12)

the same as we found in the analysis of series-series connection (Figure 1):

70665.472_1max__8 =ssfRP [W],

7677.476_3max__6 =ssfRP [W]

when splitting of frequency occurs. Variation of XAB0 in respect of frequency is presented in figure 3.

Figure 3. The variation with frequency of the Thvenin

generator reactance.

The frequencies in which the reactance XAB0_f_Th_ss is equal to zero are: f1 = 1.1619 [MHz], f2 = 1.20622 [MHz], and f3 = 1.258 [MHz]. The first and the last value represent the frequencies corresponding to the maximum of power dissipated on load resistance (Figure 4): [MHz],162.1_1max_8 =ssPRf

[MHz],208.1_2max_8 =ssPRf and

[MHz]258.1_3max_8 =ssPRf while the middle frequency

is approximately equal to the resonant one f0_ss = 1.206 [MHz].

Figure 4: Power variation in respect of frequency

The powers dissipated on load resistance and the efficiencies of the transfer power to the load resistance, corresponding to the frequencies in which the power dissipated on load resistance is maximum or minimum and to the frequencies, in which the reactance XAB0_f_Th_ss is equal to zero, are presented in Table 1.


49

Table 1.

Frequency [Hz]

Power PR8_f_ss

[W]

Efficiency eta21 [%]

7max1: 0.11618296 10f = = = =

70665.472:

_1max_8

=

ssfPR

49840524.0:

_1max_21

=

ssfeta

7max2: 0.12079818 10f = = = =

93815.268:

_2max_8

=

ssfPR

89885055.0:

_2max_21

=

ssfeta

7max3: 0.12579765 10f = = = =

7677.476:

_3max_8

=

ssfPR

50163736.0:

_3max_21

=

ssfeta

70 : 0.12065435 10f = = = = 74883.267:

_0_8

=

ssfPR

82886308.0:

_0_21

=

ssfeta

71011618869.0:

___0_01

=

ssThfXABf

56183.474:

__01_8

=

ssThfPR

49893812.0:

_01_21

=

ssfeta

71012062209.0:

___0_02

=

ssThfXABf

96340.269:

__02_8

=

ssThfPR

8287361.0:

_02_21

=

ssfeta

03_ 0 _ _ _

7: 0.1258037 10

f XAB f Th ss

= = = =

98163.472:

__03_8

=

ssThfPR

50119274.0:

_03_21

=

ssfeta

3. CONCLUSION

In the case of the series-series configuration we can remark that at the resonance frequency the maximum power transferred to the load resistance depends of the following circuit parameters: C1, C2, L3, L4, R5, R6, R7 and of the input voltage E9. The resonance frequencies at which the reactance XAB0_f_Th_ss is equal to zero are identically with the frequencies which correspond to the extreme points of the power delivered to the load resistance R8. Using the SYMNAP (SYmbolic Modified Nodal Analysis Program), the equivalent voltage generator, in respect of the terminals o o in full symbolic form, can be computed for any resonator configuration. The full symbolic form of all parameters of the equivalent voltage generator allows the optimization of the expressions of these parameters in respect of all the circuit parameters and the frequency as well. The frequencies in which the reactance XAB0_f_Th_ss is equal to zero are: f1 = 1.1619 [MHz], f2 = 1.20622 [MHz], and f3 = 1.258 [MHz]. The first and the last frequencies in which the reactance XAB0_f_Th_ss is equal to zero are very closed with the frequencies corresponding to the maximum of power dissipated on load resistance. ACKNOWLEDGMENT

This work was supported by CNCSIS - UEFISCDI, project number 678/2009 PNII - IDEI code 539/2008, and by the Sectoral Operational Programme Human Resources Development 2007-2013 of the Romanian Ministry of Labour, Family and Social Protection through the Financial Agreement POSDRU/6/1.5/S/19 (ID: 76909).

REFERENCES

[1] D. Schneider, Electrons Unplugged: Wireless power

at a distance is still far away, IEEE Spectrum Magazine, May 2010, pp. 35-39.

[2] R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, Coupled-mode theory for general free-space resonant scattering of waves, Physical Review A 75, 2007.

[3] Andr Kurs et all, Wireless power transfer via strongly coupled magnetic resonances Science Express, Vol. 317. no. 5834, pp. 83 - 86, June 2007.

[4] Andr Kurs, Power transfer through strongly coupled resonances, thesis for Master of Science in Physics under the supervision of Marin Soljai, September 2007.

[5] Robert A. Moffatt, Wireless Transfer of electric power, thesis for Bachelor of Science in Physics under the supervision of Marin Soljai, June 2009.

[6] Witricity Corporation at http://www.witricity.com/index.html.

[7] T. Imura, H. Okabe, and Y. Hori, Basic Experimental Study on Helical Antennas of Wireless Power

Transfer for Electric Vehicles by using Magnetic

Resonant Couplings, Proceedings of Vehicle Power and Propulsion Conference, September 2009, IEEE Xplore, 978-1-4244-2601-4/010/2010, pp. 936-940.

[8] M.K.Watfa, H. Al-Hassanieh, and S. Salmen, The Road to Immortal Sensor Nodes, Proceedings of International Conference on Intelligent Sensors, Sensor Networks and Information Processing, December 2008, pp. 523-528.

[9] F. Zhang, X. Liu, S.A. Hackworth, R.j. Sclabassi, and M. Sun, In Vitro and In Vivo Studies on Wireless Powering of Medical Sensors and Implantable

Devices, Proceedings of Life Science Systems and Applications Workshop, April 2009, IEEE Xplore, 978-1-4244-4293-5/09/2009, pp. 84-87.

[10] R. Bhutkar and S. Sapre, Wireless Energy Transfer using Magnetic Resonance, Second International Conference on Computer and Electrical Engineering, December 2009.

[11] A. Karalis, J.D. Joannopoulos, and M. Soljacic, Efficient wireless non-radiative mid-range energy

transfer, Annals of Physics, Vol. 323, pp. 34-48, January 2008.

[12] H. A. Haus, Microwaves and Fields in Optoelectronics, Pretince-Hall, Inc., Englewood Cliffs, New Jersey 07632, SSBN: 0-13-946053-5, 1984.

[13] H.A. Haus and W. Huang, Coupled-Mode Theory, Proc of the IEEE, Vol. 79, No. 10, October 1991, pp. 1505-1518.

[14] R. E. Hamam, A. Karalis, J.D. Joannopoulos, and M. Soljacic, Coupled-mode theory for general free-space resonant scattering of waves, Physical Review A 75, 053801, May 2007.

[15] A.A. Barybin, V. A. Dmitriev, Modern Electrodynamics and Coupled-Mode Theory:

Application to Guided-Wave Optics, Published by


50

Rinton Press, Princeton, New Jersey 08540, ISBN: 1-58949-007-X, 2002.

[16] M. Iordache, Lucia Dumitriu, D. Delion, Automatic Formulation of Symbolic State Equations for

Analog Circuits with Degeneracies, Proceedings of 6th International Workshop on Symbolic Methods and Applications in Circuit Design, SMACD 2000, Instituto Superior Tcnico, Lisbon, Portugal, October 12-13, 2000, pp. 65-72.

[17] M. Iordache, Lucia Dumitriu, L. Mandache, Time-Domain Modified Nodal Analysis for Large-Scale

Analog Circuits, Revue Roum. Sci. Techn.- lectrotechn. et nerg., Bucarest, Tome 48, No. 2-3, Bucarest , 2003, pp. 257-268, RM-ISSN 0035-4066.

[18] M. Iordache, Lucia Dumitriu, Computer Formulation of Symbolic State Equations for

Analog Nonlinear Circuits with Excess Elements, Proceedings of 2000 International Symposium on Nonlinear Theory and its Applications, NOLTA2000, Volume 1, TU Dresden Conference

Center, Dresden, Germany, September 17-21, 2000, pp. 355-358.

[19] M. Iordache, Lucia Dumitriu, Efficient Decomposition Techniques for Symbolic Analysis

of Large Scale Analog Circuits by State Variable

Method, Analog Circuits and Signal Processing, Kluwer, Vol. 40, No. 3, September 2004, Kluwer Academic Publishers, pp.235-253, Springer Netherlands ISSN: 0925-1030 (Paper) 1573-1979 (Online).

[20] Marin Soljai (researcher team leader), home page on MIT" http://www.mit.edu/~soljacic/wireless_power.html.

[21] MIT Prof. Marin Soljacic Wireless Power Transmission page http://www.mit.edu/~soljacic/wireless_power.html.

[22] M. Iordache, Lucia Dumitriu, D. Delion, SYMNAP - SYmbolic Modified Nodal Analysis Program, User Guide, Library of the Electrical Department, Politehnica University of Bucharest, Bucharest, 2000

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