research article analysis of the coupling coefficient in inductive...

Research ArticleAnalysis of the Coupling Coefficient in Inductive EnergyTransfer Systems

Rafael Mendes Duarte and Gordana Klaric Felic

The University of Melbourne Parkville VIC 3010 Australia

Correspondence should be addressed to Rafael Mendes Duarte rmendesduartegmailcom

Received 8 January 2014 Accepted 4 June 2014 Published 17 June 2014

Academic Editor Gerard Ghibaudo

Copyright copy 2014 R Mendes Duarte and G Klaric Felic This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In wireless energy transfer systems the energy is transferred from a power source to an electrical load without the need of physicalconnections In this scope inductive links have been widely studied as a way of implementing these systems Although highefficiency can be achieved when the system is operating in a static state it can drastically decrease if changes in the relative positionand in the coupling coefficient between the coils occur In this paper we analyze the coupling coefficient as a function of the distancebetween two planar and coaxial coils in wireless energy transfer systems A simple equation is derived from Neumannrsquos equationfor mutual inductance which is then used to calculate the coupling coefficient The coupling coefficient is computed using CSTMicrowave Studio and compared to calculation and experimental results for two coils with an excitation signal of up to 10MHzThe results showed that the equation presents good accuracy for geometric parameters that do not lead the solution of the ellipticintegral of the first kind to infinity

1 Introduction

Wireless power systems transfer the energy without wireconnection between a source and a loadThis provides severaladvantages specially for applications involving implantabledevices [1 2] or devices that need to be hermetically sealedduring operation [3] For the first the use of batteries presentsissues in terms of periodic replacement which may requirecomplex surgical procedures For the latter a connectionthrough wires becomes unfeasible Among the many classesof wireless systems energy transfer via inductive link hasbeen widely studied due to high efficiency and high transmis-sion power capability [4]

In inductive coupled systems a power amplifier suppliesan alternative current to the primary coil which producesa magnetic field at the secondary coil The produced fieldinduces a current and voltage at the secondary branchwhich can be rectified and supplied to a load An importantparameter for inductive links is the coupling coefficient119896 which measures how much power from the generatedelectromagnetic (EM) field is induced in the secondary coilThe coupling coefficient is related to the reflected impedance

from the secondary on the primary sideTherefore variationson 119896 change the output impedance of the power amplifierremoving it from its optimum operation condition anddecreasing its efficiency [5]

Previous studies on the magnetic coupling suggest thatthe coupling coefficient can be affected mostly due to vari-ations in the distance between the coils [6] relative anglebetween the coils and magnetic properties of the materialsurrounding the coils This is particularly important forsystemswhere the two coils are not in a fixed relative positionFor such systems the efficiency can be seriously compro-mised due to inefficient operation of the power amplifierHence analyzing the behavior of the coupling factor as afunction of these critical parameters can help to maintainthe system efficiency stable as the coupling between the coilsvaries

Numerical simulations provide highly accurate resultsbut require long computational time and extensive memoryusage Although the analytical methods based on the closedform expressions lack high accuracy they offer fast solutionsand explicit control over the design parameters

Hindawi Publishing CorporationActive and Passive Electronic ComponentsVolume 2014 Article ID 951624 6 pageshttpdxdoiorg1011552014951624

2 Active and Passive Electronic Components

Poweramplifier

Loadk

L2L1

Zin

Figure 1 Inductive link schematic

In this paper we derive a simple formula for the mutualinductance as a function of the distance between two planarand coaxial coils The equation is obtained from Neumannrsquosequation for the mutual inductance using power series asapproximated solutions of the elliptic integrals which areusually computationally expensive to obtain We also givetheoretical insight about the correlation between magneticfield distribution and coupling coefficient The presentedequation is compared to simulation and experimental resultsof two designed coils with excitation signals at frequencies ofup to 10MHz

2 Inductive Link Systems

In Figure 1 a simple diagram of an inductive link is shownA primary coil is excited by a power amplifier (PA) usuallyoperating in class E due to its high efficiency Power istransmitted to the load via magnetic coupling betweeninductors 119871

1and 119871

2 The system efficiency is defined as

the ratio between the power dissipated on the load and thepower supplied to the power amplifier In addition to lossesin the inductors due to finite resistance of the coils the PAefficiency plays a major role in degrading overall efficiency

In order to design the PA the impedance seen at itsoutput (119885in in Figure 1) must be taken into account In mostdesigns the inductance of the primary coil is used as acomponent for the output filter of the power amplifier whilethe output resistance is considered the load of the amplifierThe values of these output components are tightly relatedto the amplifier efficiency once optimum operation state isachieved for determined component values [5] Thereforevariations on the output impedance can significantly decreasethe amplifier efficiency and as a consequence the systemefficiency

For a generic load119885119871connected to the secondary coil the

impedance 119885in can be defined as

119885in = 1198771119878+ 119895119908119871

1+

11990821198722

11989511990811987121198772119878+ 119885119871

(1)

where 1198771119878

represents the series resistance of 1198711and 119872

is mutual inductance related to the coupling coefficient 119896according to

119896 =119872

radic11987111198712

(2)

where 1198711and 119871

2are the self-inductance of coils 1 and 2

Equations (1) and (2) express the dependence of theoutput impedance on the coupling coefficient of the inductivelink As the mutual inductance varies the third term of

a

b

d

Figure 2 Two coaxial coils with radii 119886 and 119887

(1) which can have an imaginary and real part varies andmodifies the impedance 119885in shown in Figure 1

3 Coupling Coefficient Calculation

Themutual inductance between two circular loops separatedby a distance 119889 and with radii 119886 and 119887 as depicted in Figure 2can be calculated using Neumannrsquos equation [7]

119872 =120583

4120587∬

cos 120598119903

119889119904 1198891199041015840 (3)

where 119889119904 and 1198891199041015840 are the incremental sections of the circularfilaments and 119903 is the distance between these two sectionswhich are defined as

119903 = radic1198862 + 1198872 + 1198892 minus 2119886 cos (120601 minus 1206011015840)

120598 = 120601 minus 1206011015840 119889119904 = 119886119889120601 119889119904 = 119887119889120601

1015840

(4)

The substitution of (4) in (3) results in

119872 =120583

4120587∬

2120587

0

119886119887 cos (120601 minus 1206011015840)1198862 + 1198872 + 1198892 minus 2119886 cos (120601 minus 1206011015840)

119889120601 1198891206011015840 (5)

The integral in (5) can be rewritten using elliptic integralsyielding

119872(119898) =2120583radic119886119887

119898[(1 minus

1198982

2)119870 (119898) minus 119864 (119898)] (6)

where 119870(119898) and 119864(119898) are the elliptic integrals of first andsecond kind respectively and119898 is defined as

119898 = radic4119886119887

(119886 + 119887)2+ 1198892

(7)

assuming values between 0 and 1

31 Solution of the Elliptic Integrals The solutions of the ellip-tic integrals of the first and second kind can be approximated[8] using (8) and (9) The approximation and the solutionobtained through numeric integration are depicted in Figures3 and 4 For low values of119898 the power series representationshows reasonable accuracyHowever119898 increases both curves

Active and Passive Electronic Components 3

04 05 06 07 08 09 10

4

8

12

16

20

24

Calculated by numeric integrationEquation (8)

m

K(m

)

Figure 3 Comparison between numerical integration and powerseries approximation for elliptic integral of the first kind

diverge from the numeric integration values For the ellipticintegral of the first kind as 119898 approaches the unity thesolution asymptotically tends to infinity much faster than thesolution calculated by numeric integration

119870 (119898) =120587

2+120587

8

1198982

1 minus 1198982 (8)

119864 (119898) =120587

2minus120587

81198982 (9)

The second term of (8) tends to infinity as119898 tends to 1

119898 = radic4119886119887

(119886 + 119887)2+ 1198892

= 1 (10)

which leads to the following relation between the coils radiusand the distance between the coils

1198892= (119886 minus 119887)

2 (11)

Hence (8) is not valid when the distance between thetwo coils approaches the value expressed by (11) In orderto avoid this limitation a high number of terms wouldbe required for the series approximation (8) making thefinal expression of the mutual inductance too complex andextensive Alternatively the solution of the elliptic integral ofthe first kind can be approximated by a logarithm function[8] However the logarithm approximation leads to issuesrelated to the signal of (6) and to an extensive resultantexpression for the mutual inductance Therefore in thispaper we only use approximations (8) and (9) analyzing itslimitations

32 Mutual Inductance Calculation The substitution of (8)and (9) in (6) yields

119872(119898) =120583120587radic119886119887

8

1198983

1 minus 1198982 (12)

104 06 0805 07 091

11

12

13

14

15

16


m

E(m

)

Figure 4 Comparison between numerical integration and powerseries approximation for elliptic integral of the second kind

++

minusminus

kI1 I2

V1 V2L1 L2

Figure 5 Circuit schematic of the inductive link

Next substituting (7) in the expression above results in theexpression for the mutual inductance as a function of thedistance between two circular coaxial loops

119872 =12058312058711988621198872

radic(119886 + 119887)2+ 1198892 [(119886 minus 119887)

2+ 1198892]

(13)

For two coils with 11989912

turns the expression can beadjusted [6] yielding

119872 =1205831205871198991119899211988621198872

radic(119886 + 119887)2+ 1198892 [(119886 minus 119887)

2+ 1198892]

(14)

which express the mutual inductance of two coils with 11989912

as a function of distance 119889 the magnetic permeability of thematerial surrounding the coils 120583 and the inner radius of thetwo coils

4 Coupling CoefficientMeasurement Technique

In this section we discuss the methodology used to infer thecoupling coefficient in simulations The circuit schematic ofthe inductive link is shown in Figure 5 and its equations canbe written as

1198811= 119895119908 (119868

11198711+1198721198682) (15)

1198811= 119895119908 (119868

21198712+1198721198681) (16)


Assuming the secondary coil is shorted (16) becomes

1198682= minus

1198721198681

1198712

(17)

Substituting (17) in (15) yields

1198811= 119895119908119868

1(1198711minus119872

1198712

) (18)

Assuming

119871119904= 1198711minus119872

1198712

(19)

equation (18) becomes

1198811= 119895119908119868

1119871119904 (20)

Hence the inductance 119871119904is the inductance measured when

the secondary coil is shorted Rearrange (19) in terms of themutual inductance

119872 = 1198712(1198711minus 119871119904) (21)

Finally combining (21) and (2) results in the expression ofthe coupling coefficient in terms of the self-inductance119871

1and

the inductance measured when the secondary coil is shorted119871119904

119896 = radic1 minus119871119904

1198711

(22)

5 Results

Simulation results were used to check the accuracy of (14)In addition experimental results were analyzed for two coilswith geometries that lead the coefficient 119898 close to unity forshort separation distances between the coils

51 Mesh Properties To evaluate the coil impedance it isnecessary to take into account the geometry and curvatureof the coils as well as the interaction between the traces ofeach turn Thus circular coils are computational expensivesince a very fine mesh is required to precisely create the EMmodel for the circular shapes of the coils In order to achievereasonable accuracy and reliable results the mesh cell sizewas decreased to a point where no further variation in theresults was observed In addition the accuracy of curvatureelements in CST was also increased until stable results wereachieved

52 Simulation In order to analyze the accuracy of (14) thecoupling coefficient between two coils with dimensions givenby Table 1 was computed using CST Parameters 119904 and 119908 arethe spacing between traces and trace width respectively ℎ isthe trace height and 119899 is the number of turns The couplingcoefficient was measured according to the method explainedin Section 4 for an excitation signal at 1 6 and 10MHz Theresults are shown in Figure 6

Table 1 Dimensions of simulated coils (in mm)

119886 ℎ 119908 119904 119899

Coil 1 115 0035 05 05 7Coil 2 115 0035 05 05 4

Table 2 Dimensions of coils used for measurements (in mm)

119886 ℎ 119908 119904 119899

235 0035 05 05 18

Equation (14) presented a reasonable accuracy comparedto simulation results specially for large separation distancebetween the coils (119889 gt 20mm) Also as (14) predicts thecoupling coefficient is not dependent on the operation fre-quency presenting small variations at different frequenciesThese small variations can be explained by proximity andskin effects which are responsible for slightly changing theimpedance of the primary and secondary coilsThe skin effectincreases proportionally to the square root of the operationfrequency increasing the impedance of the coil

53 EM Field Distribution Figure 7 shows the magnetic fielddistribution surrounding the coils at 6MHz The field linesare normal to the 119883119884 plane The coupling coefficient isdependent onhowmuchof themagnetic field flux is encircledby the secondary coil being higher if a high density oflines is projected through the secondary coil The amount ofmagnetic field lines diverging from the secondary coil can beseen as a field leakage reducing the coupling between the twocoils

As the two coils are separated apart in the 119885 directionmore field leakage is observed as the concentration of fieldlines reaching the secondary coil becomes lower

54 Measurements The limitation of (14) was also testedthrough experiments We measured the coupling coefficientof two identical coils with dimensions given in Table 2 Thesetup for the measurement is shown in Figure 8The primarycoil was connected to a port of a vector network analyzerand the 119878 parameters were measured The 119885 matrix wascalculated yielding the input impedance and therefore theinductance seen at the primary coil when the secondary coilis shorted (119871

119904in (22))

The measured and calculated results are depicted inFigure 9 The measurements were performed at an operationfrequency of 1MHz For distances above 10mm the calcu-lated results are reasonable close to the experimental onesSince the radii of both coils are the same (11) can be writtenas

1198892= 0 (23)

which means that for distances close to 0 (14) is no longervalid and good accuracy is no longer obtained as shown inFigure 9 for small distances (119889 lt 10mm)

As previously mentioned as the distance becomescloser to the condition given by (23) the second term of


0 20 4010 30 505 15 25 35 450

02

01

03

005

015

025

Calculated

k

f = 1MHzf = 6MHzf = 10MHz

d (mm)

Figure 6 Coupling coefficient as a function of distance for differentfrequencies

(Am

)

0093

00845

00761

00676

00592

00507

00423

00338

00254

00169

000845

minus303e minus 11

Figure 7 Magnetic field distribution surrounding the coils in 119884119885plane

the approximation of the elliptic integral of the first kind (8)asymptotically tends to infinity resulting in high values of thecoupling coefficient

6 Conclusions

In this paper the coupling coefficient between two coplanarand coaxial coils was analyzed as a function of the separationdistance We derived an equation for the mutual inductancefrom Neumannrsquos equation which can be used to estimatethe coupling coefficient between the coils In order to derivethe equation we used a power series approximation for thesolution of the elliptic integrals which is usually calculatedby numerical integration The derived equation presentedgood accuracy when compared to simulation results Theexperimental results demonstrated the limitation of thepresented equation as it can only be used for coil geometriesand properties that do not lead the solution of the ellipticintegral of the first kind to infinity

Figure 8 Setup for measurement of coupling coefficient

0 20102 4 6 8 12 14 16 18 22 24

CalculatedMeasuredSimulated

102

101

100

10minus1

k(lo

g)

d (mm)

Figure 9 Measured coupling coefficient at 1MHz

The analysis presented here is useful for inductive linksystems where the distance between the coils is not constantBy predicting the behavior of the coupling coefficient as afunction of the variable distance it is possible to devise asimple mechanism capable of locking the system efficiencyat a desired level despite variations in the relative position ofthe source and load

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the BrazilianNational Coun-cil for Scientific and Technological Development (CNPq) forfunding the studies of the first author in Australia Alsothey would like to thank NICTA Australiarsquos Informationand Communications Technology (ICT) Research Centre ofExcellence


References

[1] P Li and R Bashirullah ldquoA wireless power interface forrechargeable battery operated medical implantsrdquo IEEE Trans-actions on Circuits and Systems II Express Briefs vol 54 no 10pp 912ndash916 2007

[2] Q Ma M R Haider S Yuan and S K Islam ldquoPower-oscillatorbased high efficiency inductive power-link for transcutaneouspower transmissionrdquo in Proceedings of the 53rd IEEE Interna-tional Midwest Symposium on Circuits and Systems (MWSCASrsquo10) pp 537ndash540 August 2010

[3] I Muller E P Freitas A A Susin and C Pereira ldquoNamimotea low cost sensor node for wireless sensor networkrdquo in Internetof Things Smart Spaces and Next Generation Networks SAndreev S Balandin and Y Koucheryavy Eds vol 1 pp 391ndash400 Springer Berlin Germany 1st edition 2012

[4] A Kurs A Karalis R Moffatt J D Joannopoulos P Fisherand M Soljacic ldquoWireless power transfer via strongly coupledmagnetic resonancesrdquo Science vol 317 no 5834 pp 83ndash862007

[5] R L O Pinto R M Duarte F R Sousa I Muller and V JBrusamarello ldquoEfficiency modeling of class-E power oscillatorsfor wireless energy transferrdquo in Proceedings of the IEEE Interna-tional Instrumentation andMeasurement Technology Conference(I2MTC rsquo13) pp 271ndash275 May 2013

[6] S F Pichorim ldquoDesign of circular solenoid coils for maximummutual inductancerdquo in Proceedings of the14th InternationalSymposium on Biotelemetry pp 71ndash77 March 1998

[7] J C Maxwell A Treatise on Electricity and Magnetism vol 2Dover New York NY USA 1954

[8] A Russell ldquoThe magnetic field and inductance coefficients ofcircular cylindrical and helical currentsrdquo Proceedings of thePhysical Society of London vol 20 no 1 article 334 pp 476ndash506 1906

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Poweramplifier

Loadk

L2L1

Zin

Figure 1 Inductive link schematic

In this paper we derive a simple formula for the mutualinductance as a function of the distance between two planarand coaxial coils The equation is obtained from Neumannrsquosequation for the mutual inductance using power series asapproximated solutions of the elliptic integrals which areusually computationally expensive to obtain We also givetheoretical insight about the correlation between magneticfield distribution and coupling coefficient The presentedequation is compared to simulation and experimental resultsof two designed coils with excitation signals at frequencies ofup to 10MHz

2 Inductive Link Systems

In Figure 1 a simple diagram of an inductive link is shownA primary coil is excited by a power amplifier (PA) usuallyoperating in class E due to its high efficiency Power istransmitted to the load via magnetic coupling betweeninductors 119871

1and 119871

2 The system efficiency is defined as

the ratio between the power dissipated on the load and thepower supplied to the power amplifier In addition to lossesin the inductors due to finite resistance of the coils the PAefficiency plays a major role in degrading overall efficiency

In order to design the PA the impedance seen at itsoutput (119885in in Figure 1) must be taken into account In mostdesigns the inductance of the primary coil is used as acomponent for the output filter of the power amplifier whilethe output resistance is considered the load of the amplifierThe values of these output components are tightly relatedto the amplifier efficiency once optimum operation state isachieved for determined component values [5] Thereforevariations on the output impedance can significantly decreasethe amplifier efficiency and as a consequence the systemefficiency

For a generic load119885119871connected to the secondary coil the

impedance 119885in can be defined as

119885in = 1198771119878+ 119895119908119871

1+

11990821198722

11989511990811987121198772119878+ 119885119871

(1)

where 1198771119878

represents the series resistance of 1198711and 119872

is mutual inductance related to the coupling coefficient 119896according to

119896 =119872

radic11987111198712

(2)

where 1198711and 119871

2are the self-inductance of coils 1 and 2

Equations (1) and (2) express the dependence of theoutput impedance on the coupling coefficient of the inductivelink As the mutual inductance varies the third term of

a

b

d

Figure 2 Two coaxial coils with radii 119886 and 119887

(1) which can have an imaginary and real part varies andmodifies the impedance 119885in shown in Figure 1

3 Coupling Coefficient Calculation

Themutual inductance between two circular loops separatedby a distance 119889 and with radii 119886 and 119887 as depicted in Figure 2can be calculated using Neumannrsquos equation [7]

119872 =120583

4120587∬

cos 120598119903

119889119904 1198891199041015840 (3)

where 119889119904 and 1198891199041015840 are the incremental sections of the circularfilaments and 119903 is the distance between these two sectionswhich are defined as

119903 = radic1198862 + 1198872 + 1198892 minus 2119886 cos (120601 minus 1206011015840)

120598 = 120601 minus 1206011015840 119889119904 = 119886119889120601 119889119904 = 119887119889120601

1015840

(4)

The substitution of (4) in (3) results in

119872 =120583

4120587∬

2120587

0

119886119887 cos (120601 minus 1206011015840)1198862 + 1198872 + 1198892 minus 2119886 cos (120601 minus 1206011015840)

119889120601 1198891206011015840 (5)

The integral in (5) can be rewritten using elliptic integralsyielding

119872(119898) =2120583radic119886119887

119898[(1 minus

1198982

2)119870 (119898) minus 119864 (119898)] (6)

where 119870(119898) and 119864(119898) are the elliptic integrals of first andsecond kind respectively and119898 is defined as

119898 = radic4119886119887

(119886 + 119887)2+ 1198892

(7)

assuming values between 0 and 1

31 Solution of the Elliptic Integrals The solutions of the ellip-tic integrals of the first and second kind can be approximated[8] using (8) and (9) The approximation and the solutionobtained through numeric integration are depicted in Figures3 and 4 For low values of119898 the power series representationshows reasonable accuracyHowever119898 increases both curves


04 05 06 07 08 09 10

4

8

12

16

20

24


m

K(m

)



119870 (119898) =120587

2+120587

8

1198982

1 minus 1198982 (8)

119864 (119898) =120587

2minus120587

81198982 (9)


119898 = radic4119886119887

(119886 + 119887)2+ 1198892

= 1 (10)


1198892= (119886 minus 119887)

2 (11)



119872(119898) =120583120587radic119886119887

8

1198983

1 minus 1198982 (12)

104 06 0805 07 091

11

12

13

14

15

16


m

E(m

)


++

minusminus

kI1 I2

V1 V2L1 L2



119872 =12058312058711988621198872

radic(119886 + 119887)2+ 1198892 [(119886 minus 119887)

2+ 1198892]

(13)



119872 =1205831205871198991119899211988621198872

radic(119886 + 119887)2+ 1198892 [(119886 minus 119887)

2+ 1198892]

(14)





1198811= 119895119908 (119868

11198711+1198721198682) (15)

1198811= 119895119908 (119868

21198712+1198721198681) (16)



1198682= minus

1198721198681

1198712

(17)


1198811= 119895119908119868

1(1198711minus119872

1198712

) (18)

Assuming

119871119904= 1198711minus119872

1198712

(19)


1198811= 119895119908119868

1119871119904 (20)



119872 = 1198712(1198711minus 119871119904) (21)


1and


119896 = radic1 minus119871119904

1198711

(22)

5 Results





119886 ℎ 119908 119904 119899

Coil 1 115 0035 05 05 7Coil 2 115 0035 05 05 4


119886 ℎ 119908 119904 119899

235 0035 05 05 18





119904in (22))


1198892= 0 (23)




0 20 4010 30 505 15 25 35 450

02

01

03

005

015

025

Calculated

k


d (mm)


(Am

)

0093

00845

00761

00676

00592

00507

00423

00338

00254

00169

000845

minus303e minus 11



6 Conclusions



0 20102 4 6 8 12 14 16 18 22 24


102

101

100

10minus1

k(lo

g)

d (mm)





Acknowledgments



References











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










04 05 06 07 08 09 10

4

8

12

16

20

24


m

K(m

)



119870 (119898) =120587

2+120587

8

1198982

1 minus 1198982 (8)

119864 (119898) =120587

2minus120587

81198982 (9)


119898 = radic4119886119887

(119886 + 119887)2+ 1198892

= 1 (10)


1198892= (119886 minus 119887)

2 (11)



119872(119898) =120583120587radic119886119887

8

1198983

1 minus 1198982 (12)

104 06 0805 07 091

11

12

13

14

15

16


m

E(m

)


++

minusminus

kI1 I2

V1 V2L1 L2



119872 =12058312058711988621198872

radic(119886 + 119887)2+ 1198892 [(119886 minus 119887)

2+ 1198892]

(13)



119872 =1205831205871198991119899211988621198872

radic(119886 + 119887)2+ 1198892 [(119886 minus 119887)

2+ 1198892]

(14)





1198811= 119895119908 (119868

11198711+1198721198682) (15)

1198811= 119895119908 (119868

21198712+1198721198681) (16)



1198682= minus

1198721198681

1198712

(17)


1198811= 119895119908119868

1(1198711minus119872

1198712

) (18)

Assuming

119871119904= 1198711minus119872

1198712

(19)


1198811= 119895119908119868

1119871119904 (20)



119872 = 1198712(1198711minus 119871119904) (21)


1and


119896 = radic1 minus119871119904

1198711

(22)

5 Results





119886 ℎ 119908 119904 119899

Coil 1 115 0035 05 05 7Coil 2 115 0035 05 05 4


119886 ℎ 119908 119904 119899

235 0035 05 05 18





119904in (22))


1198892= 0 (23)




0 20 4010 30 505 15 25 35 450

02

01

03

005

015

025

Calculated

k


d (mm)


(Am

)

0093

00845

00761

00676

00592

00507

00423

00338

00254

00169

000845

minus303e minus 11



6 Conclusions



0 20102 4 6 8 12 14 16 18 22 24


102

101

100

10minus1

k(lo

g)

d (mm)





Acknowledgments



References











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198682= minus

1198721198681

1198712

(17)


1198811= 119895119908119868

1(1198711minus119872

1198712

) (18)

Assuming

119871119904= 1198711minus119872

1198712

(19)


1198811= 119895119908119868

1119871119904 (20)



119872 = 1198712(1198711minus 119871119904) (21)


1and


119896 = radic1 minus119871119904

1198711

(22)

5 Results





119886 ℎ 119908 119904 119899

Coil 1 115 0035 05 05 7Coil 2 115 0035 05 05 4


119886 ℎ 119908 119904 119899

235 0035 05 05 18





119904in (22))


1198892= 0 (23)




0 20 4010 30 505 15 25 35 450

02

01

03

005

015

025

Calculated

k


d (mm)


(Am

)

0093

00845

00761

00676

00592

00507

00423

00338

00254

00169

000845

minus303e minus 11



6 Conclusions



0 20102 4 6 8 12 14 16 18 22 24


102

101

100

10minus1

k(lo

g)

d (mm)





Acknowledgments



References











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 4010 30 505 15 25 35 450

02

01

03

005

015

025

Calculated

k


d (mm)


(Am

)

0093

00845

00761

00676

00592

00507

00423

00338

00254

00169

000845

minus303e minus 11



6 Conclusions



0 20102 4 6 8 12 14 16 18 22 24


102

101

100

10minus1

k(lo

g)

d (mm)





Acknowledgments



References











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










References











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article analysis of the coupling coefficient in inductive...

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