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Research ArticleModeling and Simulation of Linear and NonlinearMEMS Scale Electromagnetic Energy Harvesters for RandomVibration Environments

Farid Khan12 Boris Stoeber13 and Farrokh Sassani1

1 Department of Mechanical Engineering The University of British Columbia Vancouver BC Canada V6T 1Z42 Institute of Mechatronics Engineering University of Engineering and Technology Peshawar Pakistan3Department of Electrical and Computer Engineering The University of British Columbia Vancouver BC Canada V6T 1Z4

Correspondence should be addressed to Farid Khan dr farid khannwfpuetedupk

Received 26 August 2013 Accepted 10 November 2013 Published 30 January 2014

Academic Editors K-C Liu G Nikas S J Rothberg and B F Yousif

Copyright copy 2014 Farid Khan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The simulation results for electromagnetic energy harvesters (EMEHs) under broad band stationary Gaussian random excitationsindicate the importance of both a high transformation factor and a highmechanical quality factor to achieve favourablemean powermean square load voltage and output spectral density The optimum load is different for random vibrations and for sinusoidalvibration Reducing the total damping ratio under band-limited random excitation yields a higher mean square load voltageReduced bandwidth resulting from decreased mechanical damping can be compensated by increasing the electrical damping(transformation factor) leading to a higher mean square load voltage and power Nonlinear EMEHs with a Duffing spring and withlinear plus cubic damping are modeled using the method of statistical linearizationThese nonlinear EMEHs exhibit approximatelylinear behaviour under low levels of broadband stationary Gaussian random vibration however at higher levels of such excitationthe central (resonant) frequency of the spectral density of the output voltage shifts due to the increased nonlinear stiffness andthe bandwidth broadens slightly Nonlinear EMEHs exhibit lower maximum output voltage and central frequency of the spectraldensity with nonlinear damping compared to linear damping Stronger nonlinear damping yields broader bandwidths at stableresonant frequency

1 Introduction

The growing demand for autonomous and self-poweredsensors [1] has resulted in immense interest in harvestingenergy from the environment Similar to other energy har-vesting techniques (solar acoustic thermal or wind) [1 2]harvesting energy from ambient mechanical vibrations [3]with piezoelectric [4] electrostatic [5] and electromagnetic[6] energy harvesters has gained increasing interest in recentyears Mechanical vibrations are abundant in the environ-ment in the form of machine vibration [7] and the vibrationof household and office appliances [8] These sources havesufficient vibration levels to generate power to run ultralowpower (ULP) sensors [9] and ULP electronic circuitryhowever the frequency content of these vibrations is spreadover a wide range

Most of the developed linear and nonlinear resonantenergy harvesters have been tested and characterized underharmonic excitations however real environmental vibrationsdo not contain one steady single frequency but the vibrationis rather distributed over a broad band of frequencies andis random in nature The power spectral density (PSD) ofthe acceleration along the tangential direction of a car tire ata speed of 50 kmh for example has a rich energy contentin a broad band from 5Hz to 1 kHz [10] The vibration ofa car driven on a highway at about 105 kmh ranges from 1to 500Hz [11] The vibration levels of household appliancesreported in [12] also cover a broad bandfrom 1 to 500Hz

The models developed to predict the performance ofmicrofabricated linear resonant energy harvesters underharmonic excitation [13ndash19] are not suitable to estimatethe performance of the same devices when subjected to a

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 742580 15 pageshttpdxdoiorg1011552014742580

2 The Scientific World Journal

narrow or a broad band of random vibration A model forpiezoelectric energy harvesters under broad band randomvibration has been developed by Adhikari et al in [20]where they assume the ambient base excitation as a stationaryGaussian white noise with constant spectral density (SD)over the considered frequency range A circuit simulatorthe Simulation Program with Integrated Circuit Emphasis(SPICE) is used in [21 22] to study an energy harvesterunder broad band random vibrations With the same SPICEtechnique an electrostatic energy harvester has been simu-lated for input acceleration spectral densities of 5 times 10minus5 and5 times 10

minus4 g2Hz [21] A two-port transducer model developedfor performance tracking of linear electromechanical energyharvesters under randombroad band excitation [23] has beenextended for the analysis of linear and nonlinear piezoelectricand electrostatic harvesters excited by broad band andnarrowband random vibrations [24] Simulations of the harvesteroutput power proof mass displacement and optimum loadhave been performed under broad band Gaussian whitenoise and band limited noise excitation The author [23] hassuggested amappingmethod to extend themodel applicationto electromagnetic energy harvesters

This paper presents the analytical modeling and simula-tion results for linear and nonlinear resonant electromagneticenergy harvesters (EMEHs) under broad band and narrowband excitations The models are parameterized such thatthey are also applicable to other types of linear and nonlinearresonant EMEHs Resonant EMEHs with moving magnetor moving coil architecture with wound coil or planar coiland with uniform magnetic field or nonuniform magneticfield configuration all can be investigated The results of thiswork can be utilized for design and performance estimationof MEMS scale linear and nonlinear EMEHs under randomvibrations For broad band and narrow band random excita-tions spectral densities (SDs) of load voltage and load powermean square load voltage and mean power delivered to theload can be predicted for the harvester design parameterssuch as the mechanical quality factor the transformationfactor and the natural frequency Nonlinear harvesters withonly spring nonlinearity and with both spring and dampingnonlinearities have been modeled using the method ofstatistical linearization These nonlinear models are useful ininvestigating the effects of mechanical nonlinearity on theperformance and bandwidth of the harvesters when they aresubjected to random vibrations

2 Modeling

The EMEHs are seismic or inertial devices consisting of aninertial mass 119898 being the magnet or a proof mass and asuspension with the restoring spring force 119904(119911) to supportthe magnet or the coil During operation the motion ofthe inertial mass is damped by a damping force 119889() thatarises due to mechanical damping (air material and supportdamping) and electrical damping inducedwhen current flowsin the coil The linear and nonlinear EMEHs can be modeledas single degree of freedom spring-mass-damper systemswith base excitation as shown in Figure 1

m

d(z) s(z)

z(t)x(t)

y(t)

Figure 1 Lumped parameter model of an inertial electromagneticenergy harvester

For an excitation 119910(119905) the general form of the equation ofmotion for an inertial EMEH

119898 + d (z) + s (z) = minusmy (1)

depends on the relative acceleration (119905) relative velocity(119905) and relative displacement 119911(119905) between the permanentmagnet and the coil The expressions for the damping force119889() and the spring force 119904(119911) are modeled according to thephysical nature of the damping and stiffness present in theharvester Depending on the architecture and design of theEMEH and the excitation conditions both spring force anddamping force can be linear or one or both can be nonlinearThe behaviour of a linear EMEH (both 119904(119911) and d(z) arelinear) and a nonlinear EMEH (one or both 119904(119911) and d(z)are nonlinear) is different and requires separate models toinvestigate their performance under broad band or narrowband excitations

21 Harvester with Linear Stiffness and Linear Damping Inlinear EMEHs the spring force 119904(119911) = 119896119911 and the dampingforce d(z) = bTz are described by the linear spring stiffness119896 and the linear total damping coefficient 119887

119879

= 119887119898

+ 119887119890

respectively The mechanical damping coefficient 119887

119898

andthe electrical damping coefficient 119887

119890

contribute to the totaldamping of the harvester The equation of motion (1) forlinear EMEHs reduces to

119898 + 119887119879

+ 119896119911 = minus119898 119910 (2)

or

+ 2120577119879

120596119899

+ 1205962

119899

119911 = minus 119910 (3)

expressed in terms of the natural frequency 120596119899

and the totaldamping ratio 120577

119879

of the systemThe complex frequency response of the system

119867(119894120596) =120596

(1205962

119899

minus 1205962

) 119894 minus 2120577119879

120596119899

120596(4)

The Scientific World Journal 3

is obtained using Fourier analysis by letting 119910(119905) = 119860119890119894120596119905 and

(119905) = 119880119890119894120596119905 in (3)

The magnitude of the complex frequency response

|119867 (119894120596)| =(120596120596119899

)

120596119899

radic[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

(5)

and the SD of the base acceleration 119878119860

(120596) yield the SD of therelative velocity

119878119880

(120596) = |119867 (119894120596)|2

119878119860

(120596) (6)

The open-circuit voltage induced in EMEHs [17]

119881119866

(119905) = 119866 (119905) (7)

across the coil is directly proportional to the transformationfactor 119866 The transformation factor 119866 describes the couplingbetween the mechanical and electrical energy domains ofthe EMEH and greatly influences the energy conversionbetween these two domains For EMEHs that have a uniformmagnetic field perpendicular to the coil displacement [13] thetransformation factor

119866 = 119861119871 (8)

results from the uniform magnetic flux density 119861 and theeffective length of the coil 119871

In EMEHswith nonuniformmagnetic field configuration[17] where the coil moves in the magnetic field direction thetransformation factor

119866 = 119878d119861119911

d119911(9)

depends on the gradient d119861119911

d119911 of the normal component ofthe magnetic flux density 119861

119911

and the area sum 119878 of the coilturns

An EMEHwith the coil resistance119877119862

delivers a voltage of

119881119871

(119905) = (119877119871

119877119871

+ 119877119862

)119866 (119905) (10)

to the load resistance 119877119871

connected to the device UsingFourier analysis the load voltage in the frequency domain

119881119871

(120596) = (119877119871

119877119871

+ 119877119862

)119866 |119867 (119894120596)| 119860 (120596) (11)

contains the complex frequency response

1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816 = (

119877119871

119877119871

+ 119877119862

)119866 |119867 (119894120596)| (12)

When the EMEH is subjected to a broad band randomvibration having 119878

119860

the SD of the base acceleration and theSD of the load voltage

119878119881119871(120596) =

1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816

2

119878119860

(120596) (13)

can be expressed in the parameters of the system as

119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119878119860

(120596)

(14)

If the excitation is a stationary Gaussian random processwith zero mean the response of the system will also be astationary Gaussian with a zero mean [25] The mean squarevalue of the load voltage

1198812

119871

= int

infin

minusinfin

119878119881119871(120596) d120596

= (119877119871

119877119871

+ 119877119862

)

2

1198662

int

infin

minusinfin

|119867 (119894120596)|2

119878119860

(120596) d120596(15)

yields the average power delivered to the load resistance [24]

119875119871

=1198812

119871

119877119871

=1

119877119871

int

infin

minusinfin

119878119881119871(120596) d120596 = int

infin

minusinfin

SPL(120596) d120596 (16)

The SD of the power delivered to the load becomes

119878119875119871(120596) =

1

119877119871

119878119881119871(120596) =

1

119877119871

(119877119871

119877119871

+ 119877119862

)

2

times 1198662

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119878119860

(120596)

(17)

211 Broad Band White Noise Excitation When the excita-tion is a stationary Gaussian white noise process the SD ofthe acceleration 119878

119860

(120596) is flat and independent of frequencySubstituting the constant 119878

119860

(120596) = 1198780

in (15) yields the meansquare load voltage

1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

int

infin

minusinfin

|119867 (119894120596)|2d120596 (18)

The integral in (18) is obtained by the method describedin [26] that results in

1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577119879

120596119899

(19)

and with (16) leads to the mean power delivered to the load

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577119879

120596119899

(20)

The total damping ratio 120577119879

= 120577119898

+ 120577119890

consists of themechanical damping ratio

120577119898

=1

2119876119898

(21)


that is expressed in terms of themechanical quality factor119876119898

of the EMEH and the electrical damping ratio

120577119890

=1198662

2119898120596119899

(119877119871

+ 119877119862

)(22)

that can be obtained from the equivalent electrical circuit forthe EMEH as described in [17]

By substituting for the total damping ratio 120577119879

using (21)and (22) (20) becomes

119875119871

= 1198981205871198780

119877119871

119877119871

+ 119877119862

1198662

119876119898

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

(23)

which is more suitable to derive the optimum power condi-tion for impedance matching Optimizing (23) with respectto 119877119871

yields the condition for optimum power transfer to theload as

119877119871opt = radic119877

2

119862

+1198662

119876119898

119877119862

119898120596119899

= radic1198772

119862

+1198662

119877119862

119887119898

(24)

Equation (24) reveals that the optimum load when anEMEH is subjected to random vibration is different from theoptimum load 119877

119871opt = 119877119862

+ 1198662

119887119898

when it is subjected tosinusoidal vibration [17]

The product 1198662119876119898

in (23) is considered as the Productof merit (POM) for energy harvesters driven by randomvibrations [24] Increasing the product 1198662119876

119898

through mod-ification of the design for an EMEH will increase the meanpower delivered to the load The mechanical quality factor119876119898

can be increased by packaging the device in vacuum[27] or by incorporating air passages in the device designthat allows air flow during operation to reduce damping [28]For EMEHs with uniform magnetic field configuration thetransformation factor 119866 can be increased by increasing themagnetic flux density 119861 andor by increasing the effectivelength 119871 of the coil within the constrained footprint of thedevice For EMEHs with nonuniform magnetic field con-figuration improving the transformation factor 119866 requiresincreasing the gradient d119861

119911


119911

andor increasing the area sum119878 of the turns of the coil within the constrained footprint ofthe device However increasing the effective coil length or thenumber of coil turns also increases the coil resistance leadingto higher electrical losses especially in MEMS scale EMEHsTherefore increasing the magnetic flux density 119861 in uniformmagnetic field configuration devices and the magnetic fluxgradient in nonuniform magnetic field devices is a preferredmethod for increasing the transformation factor

Values of the mechanical quality factor 119876119898

and of thetransformation factor 119866 for various MEMS scale EMEHsreported in the literature are summarized in Table 1 Themechanical quality factor for EMEHs ranges from58 to 2587Due to their larger number of coil turns wound coil typeEMEHs exhibit higher values for the transformation factorthat contribute to the higher values of the POM 119866

2

119876119898

incomparison to planar coil type EMEHs Since these valuesvary widely for different designs they may not be suitable

for comparing various devices However for a given devicethe variation in its product of merit in response to its designparameter can be used to optimize its performance

For simulation we used the dimensions and parameters(Table 2) of our EMEH described in [17] where the nonuni-formmagnetic field is caused by twopermanentmagnetswithremanent flux density 119861

119903

which are suspended by a planarcopper spring between two identical coils

The mean power as a function of load resistance forvarious values of1198662119876

119898

is shown in Figure 2The computationwas performed for the acceleration SD of 119878

119860

(120596) = 1198780

=

001 g2rads The simulation results verify that there is anoptimum value for the load resistance for each POM 119866

2

119876119898

and that the optimum load resistance increases as the POMis increased For higher values of the POM the curves becomeincreasingly flat beyond the optimum load resistance Thisindicates that the device will perform well even at the loadresistance higher than the optimum Figure 3 shows thedependence of the mean power on the transformation factoras a function of load resistance This corresponds to the situ-ation where the mechanical quality factor 119876

119898

for the EMEHis low and remains constant while the transformation factorvaries The curves in Figure 3 are more spiked in comparisonto those in Figure 2 where the product 1198662119876

119898

is variedThis indicates that an EMEH becomes more sensitive to loadresistance variations when its 119876

119898

is low and where only thetransformation factor is changed Further the optimum loadresistance is quite different than for the case where the POMis increased

From (14) (21) and (22) the SD of the load voltage forwhite noise base excitation becomes

119878119881119871(120596)

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

119899

)

2

times ([1 minus (120596

120596119899

)

2

]

2

+ [(1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times(120596

120596119899

)]

2

)

minus1

)

(25)

The SD of the load voltage as a function of angularfrequency is shown in Figure 4 for different values of 1198662 TheSD of the load voltage shows a significant peak in the vicinityof the natural frequency of the linear EMEHThe EMEH actsas a mechanical filter and generates power in a limited bandover the bandwidth

Δ120596 = 2120577119879

120596119899

= 2 (120577119898

+ 120577119890

) 120596119899

=120596119899

119876119898

+1198662

119898(119877119871

+ 119877119862

) (26)

A broader bandwidth for the EMEH is preferred in orderto extract vibration energy from a wider band of randomexcitation A higher transformation factor leads to wider


Table1Parameterso

fMEM

Sscalee

lectromagnetic

energy

harvesters

Coiltype

m(kg)

119865resonant(H

z)119877119862

(Ω)

119877119871

(Ω)

Mechqu

ality

factor119876119898

Electqu

ality

factor119876119890

Totalqualityfactor119876119879

G(Tm)

1198662

119876119898

(T2 m

2 )Re

ferences

Wou

nd

428times10minus3

1311

1811752

653

a3118

b114

252

[29]

25times10minus3

84365

3298

8a03821

b[29]

38times10minus3

948

1227

2587c

2461a

03322

b2855

[30]

044times10minus3

350

93100

216

1120

181

041

b3631

[31]

0028times10minus3

95k

100

164

[31]

521

100

200

232

243

119[6]

102times10minus3

502323

210ndash

60604

[32]

1530

443

[32]

0028times10minus3

808

k112

26[33]

208

1218

[34]

106

1413

[34]

Planar

4036times10minus3

248

100

100

2336

[19]

00304times10minus3

100

24

162

c40323

a79

4c0017b

0005

[35]

0014times10minus3

9837

k55

164

[33]

054times10minus3

60110

110485

c234533

a00691

b0232

[33]

3139

136

120

15times10minus3

000

03[27]

221d

207d

[27]

093times10minus3

371

75100

57

583

c0075

0033

[17]

a Calculatedusingequatio

n119876

119890

=12120577

119890

b C

alculatedusingequatio

n119866=radic(119877

119871

+119877

119862

)(2120587119898119865119876

119890

)

c Calculatedusingequatio

n119876

119898

=12120577

119898

=2120587119898119865119887

119898

d D

etermined

from

testing

invacuum


Table 2 Dimensions and parameters of the EMEH prototype [17]

Description ValueDevice size 12mm times 12mm times 7mmMagnet (NdFeB) 13 TMass of each magnet 0465 gSquare spiral coil envelop 8mm times 8mmResistance of coil 119877

119862

75ΩMechanical quality factor 119876

119898

57Resonant frequency 119865resonant 371HzTransformation factor 119866 0075 Tm

bandwidth for the device However increasing the transfor-mation factor by using a larger number of turns for the coilwithin a constrained area is undesirable as this increases coilresistance that leads to power loss and negatively affects thebandwidth As seen in Figure 2 it is more significant forthe EMEH subjected to broad band vibration to optimizeboth 119876

119898

and 1198662 however increasing the mechanical qualityfactor 119876

119898

adversely affects the bandwidth of the deviceThis conflicting situation can be resolved by increasing thetransformation factor through modifications to the magneticflux density

Figure 5 shows the bandwidth of a linear EMEH as afunction of the load resistance for several values of 1198662 and119876119898

= 57 Energy harvesters with a large transformation fac-tor exhibit broader bandwidths that however drop sharplyas the load resistance is increased At a higher load resistancethe contribution due to the transformation factor term in(26) is minimal and the device bandwidth is controlled bythe dominant mechanical quality factor term However forEMEHs with a small transformation factor the contributiondue to the transformation factor term in (26) is negligible andthe bandwidth becomes independent of the load resistance asevident in Figure 5

The maximum value of the SD of the load voltage

1198781198811198711

= 119878119881119871(120596 = 120596

119899

)

=1198772

119871

119877119871

+ 119877119862

1198981198780

1198662

119876119898

120596119899

(119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

)

(27)

occurs at resonance and likewise the mean power alsodepends on the POM Increasing the POM for the EMEHwilllead to an increase in the peak value of the SD of the loadvoltage

The SD of the power as a function of angular frequencyand load resistance from (21) (22) and (17) is shown inFigure 6 Similar to the SD of the load voltage it shows anarrow peak in the vicinity of the natural frequency and atthe optimum load resistance

212 Band-Limited White Noise Excitation When the linearEMEH is excited by a stationary band-limitedGaussian whitenoise 119878

119860

(120596) = 1198780

between the angular frequency limits1205961

and

0

02

04

06

08

1

12

Mea

n po

wer

(W)

0 50 100 150 200Load resistance (Ohms)

mG2Q = 1T2 m2

G2Qm = 10T2 m2

G2Qm = 100T2 m2

G2Qm = 500T2 m2

G2Qm = 750T2 m2

G2Qm = 1000T2 m2

times10minus4

Figure 2 Mean power as a function of load resistance for differentvalues of 1198662119876

119898

002040608

1121416

Mea

n po

wer

(W)

0 20 40 60 80 100 120 140 160 180 200Load resistance (Ohms)

times10minus5

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

Figure 3 Mean power as a function of load resistance for differentvalues of 1198662 for 119876

119898

= 57

1205962

the power spectral density of the load voltage is given by(25) for 120596

1

le |120596| le 1205962

and is zero elsewhereThe SD of the load voltage for various values of 1198662 for

an EMEH excited by a band-limited random vibration from1205961

= 1640 rads to 1205962

= 3022 rads is shown in Figure 7The SD of the load voltage under band-limited excitation ismaximum in the vicinity of the natural frequency similar tothat of a broad band excitation except that it is only nonzeroover the frequency band of the input excitation

Under band-limited Gaussian white noise random exci-tation the mean square value of the load voltage

1198812

119871

= 1198780

[int

minus1205961

minus1205962

1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816

2d120596 + int1205962

1205961

1003816100381610038161003816HV (i120596)1003816100381610038161003816

2d120596] (28)


0

05

1

15

0 500 1000 1500 2000 2500 3000 3500 4000 4500Angular frequency (rads)

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f lo

ad v

olta

ge (V

rms2

rad

s)

Figure 4 SD of the load voltage as a function of angular frequency for various values of 1198662 119876119898

= 57

220240260280300320340360380

Band

wid

th (r

ads

)


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

Figure 5 Linear EMEH bandwidth as a function of the loadresistance for different values of 1198662 and 119876

119898

= 57

when expressed in terms of the total damping ratio and thefrequency ratio

1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times [

[

int

minus1205961

minus1205962

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596

+int

1205962

1205961

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596]

]

(29)

contains incomplete integrals that can be obtained by usingthemethod of partial fraction expansion [26] or can be found

0

2

4

6

022002300

240050 100 150 200

times10minus7

Load resistance (Ohm)Angular frequency (rads)

SD o

f pow

er (W

)

Figure 6 SD of the power as a function of angular frequency andload resistance for 119876

119898

= 57 and 119866 = 01Tm

with indefinite integral tables (eg byG Petit Bois 1961) [36]Equation (29) can be written in a more compact form

1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1205871198662

1198780

2120577119879

120596119899

[Γ(1205962

120596119899

120577119879

) minus Γ(1205961

120596119899

120577119879

)]

(30)

where the integral factor Γ [36] can be expressed in terms ofthe frequency ratio and the total damping ratio as

Γ(120596

120596119899

120577119879

) =1

120587arc tan

2120577119879

(120596120596119899

)

1 minus (120596120596119899

)2

minus120577119879

2120587radic1 minus 120577119879

2

times ln1 + (120596120596

119899

)2

+ 2radic1 minus 1205772

119879

(120596120596119899

)

1 + (120596120596119899

)2

minus 2radic1 minus 1205772

119879

(120596120596119899

)

(31)

In (30) the terms in front of the bracket describe themeansquare load voltage (variance) of the harvester due to broadband Gaussian white noise excitation in (19) The integralfactor Γ in the brackets is the correction factor when theexcitation is band-limited For broad band Gaussian white



0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)

Figure 7 SD of the load voltage as a function of angular frequencyfor various values of 1198662 at band-limited random excitation

120577T = 002

120577T = 006

120577T = 01

Inte

gral

fact

orΓ

Frequency ratio (120596120596n)

0

02

04

06

08

1

0 05 1 15 2

Figure 8 Integral factor for mean square load voltage of an EMEHsubjected to band-limited Gaussian white noise

noise excitation the value of the integral factor Γ(infin 120577119879

) minus

Γ(0 120577119879

) is 1 whereas for band-limited excitation it is alwaysless than 1

For three values of the total damping ratio 120577119879

= 120577119898

+

120577119890

the integral factor Γ is shown in Figure 8 The factor Γincreases monotonically as a function of the frequency ratio120596120596119899

with values residing between 0 and 1 Higher valuesof the mean square load voltage (or correction factor in thebrackets) in (30) require lower values of the total dampingfactor The electrical damping ratio 120577

119890

(or proportionally 1198662)needs to be as high as possible for high power generationtherefore for smaller values of the total damping ratiothe mechanical quality factor should be increased and theassociated reduction in the bandwidth of the device shouldbe compensated by increasing the transformation factor

3 Harvesters with Nonlinear Stiffness

For a nonlinear EMEH with linear damping force 119889() = 119887119879

and nonlinear spring force 119904(119911) = 119896119911 + 120578119896119873(119911) the generalequation of motion (1) of the harvester becomes

119898 + 119887119879

+ [119896119911 + 120578119896119873 (119911)] = minus119898 119910 (32)

in which the nonlinear spring force comprises of a linearstiffness component 119896119911 and a nonlinear stiffness component119896120578119873(119911)The scaling factor 120578 and the nonlinear function119873(119911)represent the nonlinearity of the stiffness of the harvester Foran EMEH with a symmetric suspension spring the potentialenergy is a symmetric (even) function of 119911 and that requiresthe spring force to be an antisymmetric polynomial (oddfunction) of 119911 The nonlinear function 119873(119911) is thereforea polynomial with only odd terms The scaling factor 120578represents the magnitude of the nonlinearity of the spring

A nonlinear spring force that is common in EMEHs witha polymeric membrane as the restoring member [28] can bemodeled to good approximation by a Duffing spring with thenonlinear spring force 119904(119911) = 119896119911 + 119896120578119911

3 By substituting for119904(119911) and expressing in terms of the linear natural frequency120596119899

and the total damping ratio 120577119879

(32) results in

+ 2120577119879

120596119899

+ 1205962

119899

(119911 + 1205781199113

) = minus 119910 (33)

For a stationary Gaussian random excitation with zeromean the response of the harvester will also be stationaryGaussian with zero mean The solution of (33) can beobtained by the method of statistical linearization [37ndash40]The replacement of the nonlinear component 1205962

119899

(119911 + 1205781199113

) byan equivalent linear component 1205962eq119911 yields the equation ofmotion of an equivalent linear energy harvester

+ 2120577eq119879

120596eq + 1205962

eq119911 = minus 119910 (34)

that depends on the equivalent damping ratio 120577eq119879

=

(120596119899

120596eq)120577119879 and the equivalent frequency 120596eq of an equivalentlinear EMEH To obtain an approximate solution for theresponse of the nonlinear harvester the mean square value119864[1198902

] of the error

119890 = 1205962

119899

(119911 + 1205781199113

) minus 1205962

eq119911 (35)

which would be produced by representing the nonlinear har-vester by an equivalent linear harvester must be minimizedfor the square of the equivalent frequency 120596eq that is theequation

dd1205962eq

119864 [1198902

] = 0 (36)

must be satisfiedSubstituting (35) into (36) and differentiating the result-

ing equation yields the expression for the equivalent fre-quency

1205962

eq = 1205962

119899

119864 [119911119911 (1 + 1205781199112

)]

1205902

119911

(37)


in terms of the standard deviation 120590119911

of the relative displace-ment 119911(119905) Using the method described in [37] reduces (37)to a much simpler form

1205962

eq = 1205962

119899

(1 + 31205781205902

119911

) (38)

For a Gaussian white noise random excitation the vari-ance of the relative displacement

1205902

119911

= int

infin

minusinfin

119878119911

(120596) 119889120596 = 1198780

int

infin

minusinfin

|119867 (119894120596)|2

119889120596 (39)

can be solved for the complex frequency response

119867(119894120596) =minus1

minus1205962

+ 1198942120577eq119879

120596eq120596 + 1205962

eq(40)

with the method described in [26 37] that results in

1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

2120577119879

120596119899

1205962

eq (41)

Using (38) the elimination of the equivalent frequency 120596eqfrom (41) yields a quadratic equation in 1205902

119911

31205781205904

119911

+ 1205902

119911

= 1205902

119911Lin=

1205871198780

2120577119879

1205963

119899

(42)

where 120590119911Lin

is the standard deviation of the relative displace-ment for the linear case where 120578 = 0

By substituting the positive root

1205902

119911

=

radic1 + 121205781205902

119911Linminus 1

6120578

(43)

of (42) in (38) we obtain the equivalent frequency 120596eq

1205962

eq = 1205962

119899

[[

[

1 +

radic1 + 121205781205902

119911Linminus 1

2

]]

]

(44)

that minimizes the error 119890Equations (10) and (34) yield the equation

1

119866(119877119871

+ 119877119862

119877119871

)[ddt119881119871

(119905) + 2120577eq119879

120596eq119881119871 (119905) + 1205962

eq int119881119871 (119905) dt]

= minus 119910 (119905)

(45)

which by Fourier analysis results in the frequency response ofthe harvester

119867119881

(119894120596) = (119877119871

119877119871

+ 119877119862

)119866120596

(1205962

eq minus 1205962

) 119894 minus 2120577eq119879

120596eq120596 (46)

A Gaussian white noise base excitation 119878119860

(120596) = 1198780

yieldsthe SD of the load voltage

119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

1198780

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

eq)

2


120596eq)

2

]

2

+ [(120596119899

120596eq)(

1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times (120596

120596eq)]

2

)

minus1

)

(47)

Similar to (19) the mean square load voltage of theharvester

1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(48)

and the mean power delivered to the load

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(49)

can be derived here as functions of the equivalent totaldamping ratio and the equivalent frequency

The parameters (Table 3) of our nonlinear membranetype EMEH [28] are used as reference values for simulatingnonlinear EMEHs The EMEH has a nonuniform magneticfield caused by two permanent magnets with remanent fluxdensity 119861

119903

that are suspended by a polydimethylsiloxane(PDMS) membrane between two identical coils

The SD of the load voltage for a 100Ω load at low levels ofbroad band Gaussian white noise random vibration is shownin Figure 9 The simulation is the result of (44) and (47)for a scaling factor 120578 = 5mminus2 and a mechanical qualityfactor 119876

119898

= 300 Under low levels of random vibrations thecontribution of the second term in (44) is negligible As aresult the resonant frequency is stable (not changing withincreased base excitation) showing a linear response of thedeviceTherefore the nonlinear EMEH operates in the linearregime under low levels of broad band random vibrationswhere the relative displacement of the magnets is too smallto cause a significant contribution from the nonlinear springstiffness term


Table 3 Dimensions and parameters of the nonlinear EMEHprototype [28]

Description ValueDevice size 15mm times 15mm times 10mmMagnet (NdFeB) 132 TMass of each magnet 093 gSquare spiral coil envelop 8mm times 8mmResistance of coil 101ΩLinear resonant frequency 50Hz

0

05

1

15

2

25

3

310 311 312 313 314 315 316 317 318 319 320Angular frequency (rads)

SA = 00001 g2radsSA = 00002 g2rads


times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)

Figure 9 SD of the load voltage as a function of angular frequencyfor low levels of broad bandGaussianwhite noise randomexcitationscaling factor 120578 = 5mminus2

The simulation results of the SD of the load voltage forhigher levels of broad band Gaussian white noise randomvibrations are shown in Figure 10 The maximum value ofthe load voltage spectrum increases with increasing baseacceleration moreover the central frequency of the loadvoltage SD shifts towards higher frequencies and this isattributed to the increase in the spring stiffness (resonantfrequency) of the device when it is subjected to stronger levelsof random excitation At relatively high base acceleration thelarge relative displacement of the mass invokes the nonlinearspring stiffening term and the EMEH then operates in thenonlinear regime where the resonant frequency 120596eq given by(44) increases with increasing base acceleration Moreoverin comparison to the load voltage SD under low levels ofacceleration (Figure 9) at high levels of base accelerationthe SD of the load voltage slightly broadens increasing thebandwidth of the device

4 Harvester with Nonlinear Stiffness andNonlinear Damping

For EMEH with nonlinear damping 119889() = 119887119879

+ 119887119879

120572119863()

and nonlinear stiffness 119904(119911) = 119896119911 + 119896120578119873(119911) the general formof the equation of motion (1) of the harvester becomes

119898 + [119887119879

+ 119887119879

120572119863 ()] + [119896119911 + 119896120578119873 (119911)] = minus119898 119910 (50)

0

1

2

3

4


times10minus4



SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)

Figure 10 SD of the load voltage as a function of angular frequencyfor high levels of broad band Gaussian white noise random excita-tion scaling factor 120578 = 5mminus2

A good approximation to this is obtained by assumingthe nonlinear EMEH as a Duffing oscillator with linear-plus-cubic damping The equation of motion for such a nonlinearEMEH

+ 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

) = minus 119910 (51)

contains the nonlinear damping force 119889() = 119887119879

+ 119887119879

1205723

that consists of a linear damping component 119887119879

and thenonlinear damping component 119887

119879

1205723 with 120572 as the scaling

factorWhen the excitation and response of the harvester are

both stationary Gaussian with zero mean the solution of (51)can also be obtained by the method of statistical linearization[37ndash40] The replacement of the nonlinear damping force2120577119879

120596119899

( + 1205723

) and the nonlinear spring force 1205962119899

(119911 + 1205781199113

)

by an equivalent linear damping force 120583eq = 2120577eq119879

120596eq andequivalent linear spring force 1205962eq119911 respectively yields theequation of motion of an equivalent linear energy harvester

+ 120583eq + 1205962

eq119911 = minus 119910 (52)

To obtain an approximate solution for the response of thenonlinear harvester the error

119890 = 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

) minus 120583eq minus 1205962

eq119911 (53)

resulting from this assumptionmust beminimizedThemeansquare of the error 119864[1198902] is to be minimized with respectto square of the equivalent frequency 120596eq and equivalentdamping coefficient term 120583eq that is equations

120597

1205971205962

eq119864 [1198902

] = 0

120597

120597120583eq119864 [1198902

] = 0

(54)


must be satisfied By substituting (53) into (54) two simulta-neous equations

119864 [2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [2

] minus 1205962

eq119864 [119911] = 0

119864 [1199112120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [119911] minus 1205962

eq119864 [1199112

] = 0

(55)

are obtained for the equivalent damping term 120583eq and theequivalent frequency 120596eq

For the relative displacement 119911(119905) being a stationaryGaussian random process with zero mean the substitutions119864[119911] = 0 119864[1199112] = 120590

2

119911

and 119864[2] = 1205902

119911

[37ndash39] in (55) yieldthe relation for the equivalent damping term

120583eq =119864 [2120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(56)

as a function of the standard deviation120590119911

of the relative veloc-ity (119905) as well as the relation for the equivalent frequency

1205962

eq =119864 [1199112120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(57)

as a function of the standard deviation 120590119911

of the relativedisplacement 119911(119905)The assumption of 119911(119905) and (119905) being bothGaussian yields the much simpler equations

120583eq = 119864[120597

120597[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

1205962

eq = 119864[120597

120597119911[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

(58)

which after differentiation result in the equivalent dampingterm

120583eq = 119864 [2120577119879120596119899 (1 + 31205722

)] = 120583119879

(1 + 31205721205902

119911

) (59)

as a function of the linear damping term 120583119879

= 2120577119879

120596119899

of thelinear EMEH where 120572 = 0 and the equivalent frequency

1205962

eq = 119864 [1205962

119899

(1 + 31205781199112

)] = 1205962

119899

(1 + 31205781205902

119911

) (60)

in terms of the natural frequency 120596119899

of the linear case with120578 = 0

For a Gaussian white noise random excitation the vari-ance of the relative displacement of the equivalent linearEMEH

1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

120583eq1205962

eq(61)

and the variance of the relative velocity

1205902

119911

=1205871198780

2120577eq119879

120596eq=1205871198780

120583eq(62)

can be determined with the method described in [26 37] asbefore

With the variance of the relative velocity of the linearEMEH

1205902

119911119871

=1205871198780

2120577119879120596119899

=1205871198780

120583119879

(63)

and (62) elimination of 120583eq and 120583119879

from (63) yields aquadratic equation for the variance of the relative velocity 1205902

119911

1205902

119911

(1 + 31205721205902

119911

) = 1205902

119911119871

(64)

Substitution of the positive root

1205902

119911

1205902

119911119871

=120583119879

120583eq=

radic1 + 121205721205902

119911119871

minus 1

61205721205902

119911119871

(65)

of (64) into (59) yields the relation for the equivalent dampingterm

120583eq = 120583119879[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

(66)

Similarly with the variance of the relative displacement ofthe linear EMEH

1205902

119911119871

=1205871198780

2120577119879

1205963

119899

=1205871198780

120583119879

1205962

119899

(67)


from (60) yields aquadratic equation for the variance of the relative displace-ment 1205902

119911

1205902

119911

(1 + 31205781205902

119911

)

120583eq

120583119879

= 1205902

119911119871

(68)


1205902

119911

1205902

119911119871

= (radic1 + 12119871 minus 1

6119871)120583119879

120583eq(69)

of (68) into (60) yields the relation for the equivalent naturalfrequency

1205962

eq = 1205962

119899

[1 +radic1 + 12119871 minus 1

2] (70)

where

119871 = 1205781205902

119911119871

120583119879

120583eq= 1205781205902

119911119871

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

minus1

(71)

For a Gaussian white noise base excitation 119878119860

(120596) = 1198780

the SD of the load voltage

119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

(72)


for EMEHs with combined stiffness and damping nonlin-earities can be obtained by a similar procedure described inSection 3 for EMEHs with only nonlinear stiffness

Substituting 120583eq = 2120577eq119879

120596eq and 120583119879 = 2120577119879120596119899 in (66) yieldsthe equivalent total damping ratio

120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)

Similarly (71) becomes

119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)

These allow computing the mean square load voltage

1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)

delivered to the loadFigure 11 shows the SD of the load voltage of an EMEH

with nonlinear damping and nonlinear stiffness for a 100Ωload at low levels of broad band Gaussian white randomvibration The simulation results are based on (70) (72)and (73) with a spring scaling factor 120578 = 5mminus2 and adamping scaling factor 120572 = 005 s2mminus2 In comparison tothe load voltage SD of an EMEH with linear damping andnonlinear stiffness in Figure 9 almost the same responseis obtained Under such low levels of base acceleration thelinear damping and linear stiffness terms are dominantwhereas the nonlinear damping and nonlinear stiffness termshave negligible contributions due to the small values of thestandard deviation of the relative velocity and the relativedisplacement respectively At low base accelerations thesecond term in (70) and (73) is negligible so that the


0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)

Figure 11 SD of the load voltage as a function of angular frequencyfor low levels of broad band Gaussian white random excitationspring scaling factor 120578 = 5mminus2 and damping scaling factor 120572 =

005 s2mminus2

nonlinear EMEH operates in the linear regime with a stablecentral frequency (resonant frequency) of the load voltageSD

The simulation results of EMEHswith nonlinear dampingand stiffness at relatively high levels of broad band Gaussianwhite noise random vibrations are shown in Figure 12With an increased base acceleration level the shift of thecentral frequency of the load voltage SD towards higherfrequencies indicates the operation of the device in thenonlinear regime Under these conditions the higher valuesof the standard deviations of the relative velocity and therelative displacement of the mass invoke the nonlinear effectsof the system In other words the contribution of the secondterms in (70) and (73) becomes significant

In comparison to the response of EMEHs with nonlinearstiffness only as in Figure 10 the same shift in the SDmaximum value and central frequency is evident in the caseof the fully nonlinear harvester However in Figure 12 theseshifts are less significant due to the existence of the nonlineardamping The nonlinear damping of the EMEH whichincreases as the standard deviation of the relative velocityrises does not allow the same increase in themaximum valuefor the load voltage SD and the central frequency as in caseof the EMEH with linear damping Moreover the higherdamping leads to broader bandwidths in comparison to theEMEH with nonlinear stiffness only in Figure 10

The response of the nonlinear EMEH with a largerdamping scaling factor 120572 = 5 s2mminus2 is shown in Figure 13In this case a much smaller increase in the maximum valueof the load SD is seen moreover the central frequency isalmost constant and does not change with increasing baseacceleration Broader bandwidths are obtained in compari-son to a nonlinear EMEHwith smaller damping scaling factor120572 = 005 s2mminus2 The larger value of the nonlinear dampingterm diminishes the effects of the nonlinear stiffness termuntil and unless the spring scaling factor is very large

For a nonlinear EMEH with combined nonlinear stiff-ness and damping the equivalent resonant frequency for a


002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)

Figure 12 SD of the load voltage as a function of angular frequencyfor high levels of broad band Gaussian white noise random excita-tion spring scaling factor 120578 = 5mminus2 and damping scaling factor120572 = 005 s2mminus2

002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)

Figure 13 SD of the load voltage as a function of angular frequencyfor high levels of broad band Gaussian white random excitationspring scaling factor 120578 = 5mminus2 and damping scaling factor 120572 =

5 s2mminus2

stiffness damping factor of 120578 = 5mminus2 and several valuesfor the damping scaling factor 120572 is plotted in Figure 14When the EMEH is subjected to increasing SD levels of theacceleration for smaller values of120572 the shift in the equivalentfrequency of the response is significant however this shiftdecreases as 120572 is increased For 120572 = 5 s2mminus2 or largervalues the change in the equivalent frequency is negligibleMoreover it can be seen from the plot that at lower SD levelsof the acceleration (eg at 119878

119860

= 00001 g2rads) the shift inresonant frequency is minimal even if the difference in 120572 islarge This indicates that at the excitation level equal to orless than 00001 g2rads contributions from the nonlineareffects are negligible The EMEH will be operating in thelinear regime with approximately constant resonant (central)frequency of the SD of the response

The mean power as a function of load resistance forseveral values of the transformation factor is shown in

3141

3143

3145

3147

0 002 004 006

SD of acceleration (g2rads)

Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2

Figure 14 Equivalent frequency as a function of SD of accelerationfor several values of the damping scaling factor 120572 with 120578 = 5mminus2

3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)

Load resistance (Ohms)

times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

Figure 15 Mean power as a function of load resistance for differentvalues of 1198662 with spring scaling factor 120578 = 5mminus2 and the dampingscaling factor 120572 = 5 s2mminus2 at SD = 001 g2rads

Figure 15 The plots are obtained by using (70) and (73) in(76) The computation is performed for the acceleration SDof 001 g2rads With increase in the transformation factorthe peak value of the mean power increases and the optimumload changes

5 Conclusions

Analytical models for linear and nonlinear electromagneticenergy harvesters (EMEHs) for random vibrations have beendeveloped and simulations were performed to predict thebehaviour of these harvesters under broad band randomexcitations In contrast to harmonic excitation the simulationresults have shown different output responses when linearand nonlinear EMEHs are subjected to broad band randomexcitation


For linear EMEHs under broad band stationary Gaussianrandom excitation the simulation results of the mean powermean square load voltage spectral density of the deviceoutput and the harvester bandwidth show the significance ofboth the transformation and the mechanical quality factorsFor larger values of the product of merit the mean powerbecomes less dependent on the optimum load and thedevice can be operated off its optimum load condition Theincrease of the transformation factor for EMEHs with a smallmechanical quality factor required the device to be operatedat the optimum load for better performance Moreover theoptimum load condition under random vibration is quitedifferent from that of harmonic excitations Higher valuesof the transformation factor have shown to broaden thebandwidth of the harvester and the bandwidth is dependenton the load resistance in such a case however at lowertransformation factor the bandwidth solely depends on themechanical quality factor and is independent of the loadresistance

For linear EMEHs the SD of the load voltage under band-limited random excitation is nonzero only over the inputfrequency band and is maximum in the vicinity of the naturalfrequency similar to that of broad band excitationThe meansquare load voltage depends on the frequency band of theexcitation and the total damping ratio of the harvesterHighervalues of the mean square of the load voltage require smallvalues of the total damping ratio The electrical dampingratio (transformation factor) should be as high as possiblefor high power generation therefore for smaller values ofthe total damping ratio the mechanical quality factor mustbe increased The associated reduction in the bandwidth ofthe harvester must then be compensated by increasing thetransformation factor

For nonlinear EMEHs the statistical linearizationmethodwas used for modeling under broad band random vibrationThe response of nonlinear EMEHs not only depends onthe spectral density of the base acceleration but also on thestandard deviations of the relative velocity and the relativedisplacement Under low levels of random excitation thecontribution from the nonlinear terms is negligible the linearstiffness and linear damping are dominant harvesters operatein the linear regime and the response of a nonlinear deviceis just similar to a linear EMEH When these nonlinear har-vesters are subjected to higher levels of random excitationsto invoke the nonlinear effects the simulation results haveshown that not only the maximum value of the load voltageis increased but also the central (resonant) frequency of thespectral density has been shifted towards higher frequenciesThe shift in the central frequency is attributed to the increasedstiffnessHowever this shift becomes less significant in case ofEMEHs with nonlinear damping since the nonlinear damp-ing term contributes inversely to the resonance frequencyMoreover slightly broader bandwidths are obtained in thenonlinear regime in comparison to operating in the linearregime The presence of high levels of nonlinear dampingnot only increases the bandwidth of the harvester at theexpense of decreased peak value of the load voltage spectraldensity but it also leads to a stable resonant frequency evenat relatively high levels of random excitation

Conflict of Interests

The authors declare that there is no conflict of interests

References

[1] E Sardini and M Serpelloni ldquoPassive and self-poweredautonomous sensors for remote measurementsrdquo Sensors vol 9no 2 pp 943ndash960 2009

[2] S Roundy P K Wright and J M Rabaey Energy Scavengingfor Wireless Sensor Networks with Special Focus on VibrationsKluwer-Academic Norwell Mass USA 2004

[3] S P Beeby M J Tudor and N M White ldquoEnergy harvestingvibration sources for microsystems applicationsrdquoMeasurementScience and Technology vol 17 no 12 article R175 2006

[4] S R Anton and H A Sodano ldquoA review of power harvestingusing piezoelectric materials (2003ndash2006)rdquo Smart Materialsand Structures vol 16 no 3 article R1 2007

[5] D Hoffmann B Folkmer and Y Manoli ldquoFabrication char-acterization and modelling of electrostatic micro-generatorsrdquoJournal of Micromechanics and Microengineering vol 19 no 9Article ID 094001 2009

[6] S P Beeby R N Torah M J Tudor et al ldquoA micro electro-magnetic generator for vibration energy harvestingrdquo Journal ofMicromechanics and Microengineering vol 17 no 7 article 12572007

[7] V Wowk A Brief Tutorial on Machine Vibration MachineDynamics 2005 httpwwwmachinedyncomrevisedtutorialpdf

[8] R X Gao and Y Cui ldquoVibration-based energy extraction forsensor powering design analysis and experimental evalua-tionrdquo in Smart Structures and Materials Sensors and SmartStructures Technologies for Civil Mechanical and AerospaceSystem vol 5765 of Proceedings of SPIE pp 794ndash801 San DiegoCalif USA March 2005

[9] F Khan F Sassani and B Stoeber ldquoVibration-based elec-tromagnetic energy harvesterrdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress amp Exposition(IMECE rsquo10) Vancouver Canada November 2010

[10] M Lohndorf T Kvisteroy E Westby and E HalvorsenldquoEvaluation of energy harvesting concepts for a tire pressuremonitoring systemsrdquo in Proceedings of the Power MEMS pp331ndash334 Freiburg Germany November 2007

[11] S Priya ldquoAdvances in energy harvesting using low profilepiezoelectric transducersrdquo Journal of Electroceramics vol 19 no1 pp 165ndash182 2007

[12] S Roundy P K Wright and J Rabaey ldquoA study of lowlevel vibrations as a power source for wireless sensor nodesrdquoComputer Communications vol 26 no 11 pp 1131ndash1144 2003

[13] P D Mitcheson T C Green E M Yeatman and A S HolmesldquoArchitectures for vibration-driven micropower generatorsrdquoJournal of Microelectromechanical Systems vol 13 no 3 pp429ndash440 2004

[14] N G Stephen ldquoOn energy harvesting from ambient vibrationrdquoJournal of Sound and Vibration vol 293 no 1-2 pp 409ndash4252006

[15] C B Williams and R B Yates ldquoAnalysis of a micro-electricgenerator for microsystemsrdquo Sensors and Actuators A vol 52no 1-3 pp 8ndash11 1996

[16] T Sterken K Baert C Van Hoof R Puers G Borghs andP Fiorini ldquoComparative modeling for vibration scavengersrdquo in


Proceedings of the IEEE Sensors Conference vol 3 pp 1249ndash1252Vienna Austria October 2004

[17] F Khan F Sassani and B Stoeber ldquoCopper foil-type vibration-based electromagnetic energy harvesterrdquo Journal of Microme-chanics and Microengineering vol 20 no 12 Article ID 1250062010

[18] P Wang K Tanaka S Sugiyama X Dai X Zhao and J LiuldquoA micro electromagnetic low level vibration energy harvesterbased on MEMS technologyrdquoMicrosystem Technologies vol 15no 6 pp 941ndash951 2009

[19] G Hatipoglu and H Urey ldquoFR4-based electromagnetic energyharvester for wireless sensor nodesrdquo Smart Materials andStructures vol 19 no 1 Article ID 015022 2010

[20] S Adhikari M I Friswell and D J Inman ldquoPiezoelectricenergy harvesting from broadband random vibrationsrdquo SmartMaterials and Structures vol 18 no 11 Article ID 115005 2009

[21] E Halvorsen L C J Blystad S Husa and E Westby ldquoSim-ulation of electromechanical systems driven by large randomvibrationsrdquo in Proceedings of the IEEE International Conferenceon Perspective Technologies and Methods in MEMS Design(MEMSTECH rsquo07) pp 117ndash122 Lviv-Polyana Ukraine May2007

[22] L GW Tvedt L-C J Blystad and E Halvorsen ldquoSimulation ofan electrostatic energy harvester at large amplitude narrow andwide band vibrationsrdquo in Proceedings of the IEEE Symposiumon Design Test Integration and Packaging of MEMSMOEMS(DTIP of MEMS and MOEMS rsquo08) pp 296ndash301 Nice FranceApril 2008

[23] E Halvorsen ldquoBroadband excitation of resonant energy har-vestersrdquo in Proceedings of the PowerMEMS pp 319ndash322Freiburg Germany November 2007

[24] E Halvorsen ldquoEnergy harvesters driven by broadband randomvibrationsrdquo Journal of Microelectromechanical Systems vol 17no 5 pp 1061ndash1071 2008

[25] A Dimarogonas Vibration for Engineers Prentice-Hall NewJersey NJ USA 1996

[26] SH Crandall andWDMarkRandomVibration inMechanicalSystems Academic Press New York NY USA 1963

[27] C B Williams C Shearwood M A Harradine P H Mellor TS Birch and R B Yates ldquoDevelopment of an electromagneticmicro-generatorrdquo Proceedings of the IEEE Circuits Devices andSystems vol 148 pp 337ndash342 December 2001

[28] F Khan F Sassani and B Stoeber ldquoVibration-based PDMSmembrane type electromagnetic power generator for lowvibration environmentsrdquo in Proceedings of the CSME ForumVictoria Canada June 2010

[29] C R Saha T OrsquoDonnell H Loder S Beeby and J TudorldquoOptimization of an electromagnetic energy harvesting devicerdquoIEEE Transactions on Magnetics vol 42 no 10 pp 3509ndash35112006

[30] M S M Soliman E M Abdel-Rahman E F El-Saadany andR R Mansour ldquoA wideband vibration-based energy harvesterrdquoJournal of Micromechanics and Microengineering vol 18 no 11Article ID 115021 2008

[31] S P Beeby M J Tudor R N Torah et al ldquoMacro and microscale electromagnetic kinetic energy harvesting generatorsrdquo inProceedings of the Design Test Integration and Packaging ofMEMSampMOEMS (DTIP MEMSampMOEMS rsquo06) Stresa ItalyApril 2006

[32] R Torah P Glynne-Jones M Tudor T OrsquoDonnell S Royand S Beeby ldquoSelf-powered autonomous wireless sensor node

using vibration energy harvestingrdquo Measurement Science andTechnology vol 19 no 12 Article ID 125202 2008

[33] S Kulkarni E Koukharenko R Torah et al ldquoDesign fab-rication and test of integrated micro-scale vibration-basedelectromagnetic generatorrdquo Sensors and Actuators A vol 145-146 no 1-2 pp 336ndash342 2008

[34] P Glynne-Jones M J Tudor S P Beeby and N M White ldquoAnelectromagnetic vibration-powered generator for intelligentsensor systemsrdquo Sensors and Actuators A vol 110 no 1ndash3 pp344ndash349 2004

[35] W-S Huang K-E Tzeng M-C Cheng and R-S HuangldquoA silicon MEMS micro power generator for wearable microdevicesrdquo Journal of the Chinese Institute of Engineers Transac-tions of the Chinese Institute of Engineers A vol 30 no 1 pp133ndash140 2007

[36] A Preumont Random Vibration and Spectral Analysis Kluwer-Academic Dordrecht Netherlands 1994

[37] J B Roberts and P D Spanos Random Vibration and StatisticalLinearization John Wiley amp Sons Sussex UK 1990

[38] Nigam Introduction to Random Vibrations The MIT PressLondon UK 1983

[39] F Dinca and C Teodosiu Nonlinear and Random VibrationsAcademic Press New York NY USA 1973

[40] A R Bulsara K Lindenberg andK E Shuler ldquoSpectral analysisof a nonlinear oscillator driven by random and periodic forcesI Linearized theoryrdquo Journal of Statistical Physics vol 27 no 4pp 787ndash808 1982

International Journal of

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Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



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Journal of

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



narrow or a broad band of random vibration A model forpiezoelectric energy harvesters under broad band randomvibration has been developed by Adhikari et al in [20]where they assume the ambient base excitation as a stationaryGaussian white noise with constant spectral density (SD)over the considered frequency range A circuit simulatorthe Simulation Program with Integrated Circuit Emphasis(SPICE) is used in [21 22] to study an energy harvesterunder broad band random vibrations With the same SPICEtechnique an electrostatic energy harvester has been simu-lated for input acceleration spectral densities of 5 times 10minus5 and5 times 10

minus4 g2Hz [21] A two-port transducer model developedfor performance tracking of linear electromechanical energyharvesters under randombroad band excitation [23] has beenextended for the analysis of linear and nonlinear piezoelectricand electrostatic harvesters excited by broad band andnarrowband random vibrations [24] Simulations of the harvesteroutput power proof mass displacement and optimum loadhave been performed under broad band Gaussian whitenoise and band limited noise excitation The author [23] hassuggested amappingmethod to extend themodel applicationto electromagnetic energy harvesters

This paper presents the analytical modeling and simula-tion results for linear and nonlinear resonant electromagneticenergy harvesters (EMEHs) under broad band and narrowband excitations The models are parameterized such thatthey are also applicable to other types of linear and nonlinearresonant EMEHs Resonant EMEHs with moving magnetor moving coil architecture with wound coil or planar coiland with uniform magnetic field or nonuniform magneticfield configuration all can be investigated The results of thiswork can be utilized for design and performance estimationof MEMS scale linear and nonlinear EMEHs under randomvibrations For broad band and narrow band random excita-tions spectral densities (SDs) of load voltage and load powermean square load voltage and mean power delivered to theload can be predicted for the harvester design parameterssuch as the mechanical quality factor the transformationfactor and the natural frequency Nonlinear harvesters withonly spring nonlinearity and with both spring and dampingnonlinearities have been modeled using the method ofstatistical linearization These nonlinear models are useful ininvestigating the effects of mechanical nonlinearity on theperformance and bandwidth of the harvesters when they aresubjected to random vibrations

2 Modeling

The EMEHs are seismic or inertial devices consisting of aninertial mass 119898 being the magnet or a proof mass and asuspension with the restoring spring force 119904(119911) to supportthe magnet or the coil During operation the motion ofthe inertial mass is damped by a damping force 119889() thatarises due to mechanical damping (air material and supportdamping) and electrical damping inducedwhen current flowsin the coil The linear and nonlinear EMEHs can be modeledas single degree of freedom spring-mass-damper systemswith base excitation as shown in Figure 1

m

d(z) s(z)

z(t)x(t)

y(t)

Figure 1 Lumped parameter model of an inertial electromagneticenergy harvester

For an excitation 119910(119905) the general form of the equation ofmotion for an inertial EMEH

119898 + d (z) + s (z) = minusmy (1)

depends on the relative acceleration (119905) relative velocity(119905) and relative displacement 119911(119905) between the permanentmagnet and the coil The expressions for the damping force119889() and the spring force 119904(119911) are modeled according to thephysical nature of the damping and stiffness present in theharvester Depending on the architecture and design of theEMEH and the excitation conditions both spring force anddamping force can be linear or one or both can be nonlinearThe behaviour of a linear EMEH (both 119904(119911) and d(z) arelinear) and a nonlinear EMEH (one or both 119904(119911) and d(z)are nonlinear) is different and requires separate models toinvestigate their performance under broad band or narrowband excitations

21 Harvester with Linear Stiffness and Linear Damping Inlinear EMEHs the spring force 119904(119911) = 119896119911 and the dampingforce d(z) = bTz are described by the linear spring stiffness119896 and the linear total damping coefficient 119887

119879

= 119887119898

+ 119887119890

respectively The mechanical damping coefficient 119887

119898

andthe electrical damping coefficient 119887

119890

contribute to the totaldamping of the harvester The equation of motion (1) forlinear EMEHs reduces to

119898 + 119887119879

+ 119896119911 = minus119898 119910 (2)

or

+ 2120577119879

120596119899

+ 1205962

119899

119911 = minus 119910 (3)

expressed in terms of the natural frequency 120596119899

and the totaldamping ratio 120577

119879

of the systemThe complex frequency response of the system

119867(119894120596) =120596

(1205962

119899

minus 1205962

) 119894 minus 2120577119879

120596119899

120596(4)



(119905) = 119880119890119894120596119905 in (3)


|119867 (119894120596)| =(120596120596119899

)

120596119899

radic[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

(5)



119878119880

(120596) = |119867 (119894120596)|2

119878119860

(120596) (6)


119881119866

(119905) = 119866 (119905) (7)


119866 = 119861119871 (8)



119866 = 119878d119861119911

d119911(9)



119911




119881119871

(119905) = (119877119871

119877119871

+ 119877119862

)119866 (119905) (10)



119881119871

(120596) = (119877119871

119877119871

+ 119877119862

)119866 |119867 (119894120596)| 119860 (120596) (11)


1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816 = (

119877119871

119877119871

+ 119877119862

)119866 |119867 (119894120596)| (12)


119860


119878119881119871(120596) =

1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816

2

119878119860

(120596) (13)


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119878119860

(120596)

(14)


1198812

119871

= int

infin

minusinfin

119878119881119871(120596) d120596

= (119877119871

119877119871

+ 119877119862

)

2

1198662

int

infin

minusinfin

|119867 (119894120596)|2

119878119860

(120596) d120596(15)


119875119871

=1198812

119871

119877119871

=1

119877119871

int

infin

minusinfin

119878119881119871(120596) d120596 = int

infin

minusinfin

SPL(120596) d120596 (16)


119878119875119871(120596) =

1

119877119871

119878119881119871(120596) =

1

119877119871

(119877119871

119877119871

+ 119877119862

)

2

times 1198662

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119878119860

(120596)

(17)


119860


119860

(120596) = 1198780


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

int

infin

minusinfin

|119867 (119894120596)|2d120596 (18)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577119879

120596119899

(19)


119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577119879

120596119899

(20)


= 120577119898

+ 120577119890


120577119898

=1

2119876119898

(21)




120577119890

=1198662

2119898120596119899

(119877119871

+ 119877119862

)(22)




119875119871

= 1198981205871198780

119877119871

119877119871

+ 119877119862

1198662

119876119898

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

(23)



119877119871opt = radic119877

2

119862

+1198662

119876119898

119877119862

119898120596119899

= radic1198772

119862

+1198662

119877119862

119887119898

(24)


119871opt = 119877119862

+ 1198662

119887119898


The product 1198662119876119898


119898



119911


119911




2

119876119898




119903



119898


119860

(120596) = 1198780

=


2

119876119898


119898


119898


119898



119878119881119871(120596)

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

119899

)

2


120596119899

)

2

]

2

+ [(1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times(120596

120596119899

)]

2

)

minus1

)

(25)


Δ120596 = 2120577119879

120596119899

= 2 (120577119898

+ 120577119890

) 120596119899

=120596119899

119876119898

+1198662

119898(119877119871

+ 119877119862

) (26)



Table1Parameterso

fMEM

Sscalee

lectromagnetic

energy

harvesters

Coiltype

m(kg)

119865resonant(H

z)119877119862

(Ω)

119877119871

(Ω)

Mechqu

ality

factor119876119898

Electqu

ality

factor119876119890


G(Tm)

1198662

119876119898

(T2 m

2 )Re

ferences

Wou

nd

428times10minus3

1311

1811752

653

a3118

b114

252

[29]

25times10minus3

84365

3298

8a03821

b[29]

38times10minus3

948

1227

2587c

2461a

03322

b2855

[30]

044times10minus3

350

93100

216

1120

181

041

b3631

[31]

0028times10minus3

95k

100

164

[31]

521

100

200

232

243

119[6]

102times10minus3

502323

210ndash

60604

[32]

1530

443

[32]

0028times10minus3

808

k112

26[33]

208

1218

[34]

106

1413

[34]

Planar

4036times10minus3

248

100

100

2336

[19]

00304times10minus3

100

24

162

c40323

a79

4c0017b

0005

[35]

0014times10minus3

9837

k55

164

[33]

054times10minus3

60110

110485

c234533

a00691

b0232

[33]

3139

136

120

15times10minus3

000

03[27]

221d

207d

[27]

093times10minus3

371

75100

57

583

c0075

0033

[17]


n119876

119890

=12120577

119890

b C


n119866=radic(119877

119871

+119877

119862

)(2120587119898119865119876

119890

)


n119876

119898

=12120577

119898

=2120587119898119865119887

119898

d D

etermined

from

testing

invacuum




119862


119898



119898


119898





1198781198811198711

= 119878119881119871(120596 = 120596

119899

)

=1198772

119871

119877119871

+ 119877119862

1198981198780

1198662

119876119898

120596119899

(119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

)

(27)




119860

(120596) = 1198780


and

0

02

04

06

08

1

12

Mea

n po

wer

(W)


mG2Q = 1T2 m2

G2Qm = 10T2 m2

G2Qm = 100T2 m2

G2Qm = 500T2 m2

G2Qm = 750T2 m2

G2Qm = 1000T2 m2

times10minus4


119898

002040608

1121416

Mea

n po

wer

(W)


times10minus5

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2


119898

= 57

1205962


1

le |120596| le 1205962



= 1640 rads to 1205962



1198812

119871

= 1198780

[int

minus1205961

minus1205962

1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816

2d120596 + int1205962

1205961

1003816100381610038161003816HV (i120596)1003816100381610038161003816

2d120596] (28)


0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f lo

ad v

olta

ge (V

rms2

rad

s)


= 57

220240260280300320340360380

Band

wid

th (r

ads

)


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2


119898

= 57


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times [

[

int

minus1205961

minus1205962

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596

+int

1205962

1205961

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596]

]

(29)


0

2

4

6

022002300

240050 100 150 200

times10minus7


SD o

f pow

er (W

)


119898

= 57 and 119866 = 01Tm


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1205871198662

1198780

2120577119879

120596119899

[Γ(1205962

120596119899

120577119879

) minus Γ(1205961

120596119899

120577119879

)]

(30)


Γ(120596

120596119899

120577119879

) =1

120587arc tan

2120577119879

(120596120596119899

)

1 minus (120596120596119899

)2

minus120577119879

2120587radic1 minus 120577119879

2

times ln1 + (120596120596

119899

)2


119879

(120596120596119899

)

1 + (120596120596119899

)2


119879

(120596120596119899

)

(31)




0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


120577T = 002

120577T = 006

120577T = 01

Inte

gral

fact

orΓ


0

02

04

06

08

1

0 05 1 15 2



) minus

Γ(0 120577119879



= 120577119898

+

120577119890



119890





119898 + 119887119879

+ [119896119911 + 120578119896119873 (119911)] = minus119898 119910 (32)





(32) results in

+ 2120577119879

120596119899

+ 1205962

119899

(119911 + 1205781199113

) = minus 119910 (33)


119899

(119911 + 1205781199113


+ 2120577eq119879

120596eq + 1205962

eq119911 = minus 119910 (34)


=

(120596119899


] of the error

119890 = 1205962

119899

(119911 + 1205781199113

) minus 1205962

eq119911 (35)


dd1205962eq

119864 [1198902

] = 0 (36)



1205962

eq = 1205962

119899

119864 [119911119911 (1 + 1205781199112

)]

1205902

119911

(37)




1205962

eq = 1205962

119899

(1 + 31205781205902

119911

) (38)


1205902

119911

= int

infin

minusinfin

119878119911

(120596) 119889120596 = 1198780

int

infin

minusinfin

|119867 (119894120596)|2

119889120596 (39)


119867(119894120596) =minus1

minus1205962

+ 1198942120577eq119879

120596eq120596 + 1205962

eq(40)


1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

2120577119879

120596119899

1205962

eq (41)


119911

31205781205904

119911

+ 1205902

119911

= 1205902

119911Lin=

1205871198780

2120577119879

1205963

119899

(42)

where 120590119911Lin



1205902

119911

=

radic1 + 121205781205902

119911Linminus 1

6120578

(43)


1205962

eq = 1205962

119899

[[

[

1 +

radic1 + 121205781205902

119911Linminus 1

2

]]

]

(44)


1

119866(119877119871

+ 119877119862

119877119871

)[ddt119881119871

(119905) + 2120577eq119879

120596eq119881119871 (119905) + 1205962

eq int119881119871 (119905) dt]

= minus 119910 (119905)

(45)


119867119881

(119894120596) = (119877119871

119877119871

+ 119877119862

)119866120596

(1205962

eq minus 1205962

) 119894 minus 2120577eq119879

120596eq120596 (46)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

1198780

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

eq)

2


120596eq)

2

]

2

+ [(120596119899

120596eq)(

1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times (120596

120596eq)]

2

)

minus1

)

(47)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(48)


119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(49)



119903



119898





0

05

1

15

2

25

3




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)





+ 119887119879

120572119863()


119898 + [119887119879

+ 119887119879

120572119863 ()] + [119896119911 + 119896120578119873 (119911)] = minus119898 119910 (50)

0

1

2

3

4


times10minus4



SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)



+ 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

) = minus 119910 (51)


+ 119887119879

1205723



119879




120596119899

( + 1205723


(119911 + 1205781199113

)



+ 120583eq + 1205962

eq119911 = minus 119910 (52)


119890 = 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113


eq119911 (53)


120597

1205971205962

eq119864 [1198902

] = 0

120597

120597120583eq119864 [1198902

] = 0

(54)



119864 [2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [2

] minus 1205962

eq119864 [119911] = 0

119864 [1199112120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [119911] minus 1205962

eq119864 [1199112

] = 0

(55)



2

119911

and 119864[2] = 1205902

119911


120583eq =119864 [2120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(56)



1205962

eq =119864 [1199112120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(57)



120583eq = 119864[120597

120597[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

1205962

eq = 119864[120597

120597119911[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

(58)


120583eq = 119864 [2120577119879120596119899 (1 + 31205722

)] = 120583119879

(1 + 31205721205902

119911

) (59)


= 2120577119879

120596119899


1205962

eq = 119864 [1205962

119899

(1 + 31205781199112

)] = 1205962

119899

(1 + 31205781205902

119911

) (60)




1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

120583eq1205962

eq(61)


1205902

119911

=1205871198780

2120577eq119879

120596eq=1205871198780

120583eq(62)



1205902

119911119871

=1205871198780

2120577119879120596119899

=1205871198780

120583119879

(63)



119911

1205902

119911

(1 + 31205721205902

119911

) = 1205902

119911119871

(64)


1205902

119911

1205902

119911119871

=120583119879

120583eq=

radic1 + 121205721205902

119911119871

minus 1

61205721205902

119911119871

(65)


120583eq = 120583119879[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

(66)


1205902

119911119871

=1205871198780

2120577119879

1205963

119899

=1205871198780

120583119879

1205962

119899

(67)



119911

1205902

119911

(1 + 31205781205902

119911

)

120583eq

120583119879

= 1205902

119911119871

(68)


1205902

119911

1205902

119911119871

= (radic1 + 12119871 minus 1

6119871)120583119879

120583eq(69)


1205962

eq = 1205962

119899

[1 +radic1 + 12119871 minus 1

2] (70)

where

119871 = 1205781205902

119911119871

120583119879

120583eq= 1205781205902

119911119871

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

minus1

(71)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

(72)





120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)


119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)




0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


005 s2mminus2







002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(119905) = 119880119890119894120596119905 in (3)


|119867 (119894120596)| =(120596120596119899

)

120596119899

radic[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

(5)



119878119880

(120596) = |119867 (119894120596)|2

119878119860

(120596) (6)


119881119866

(119905) = 119866 (119905) (7)


119866 = 119861119871 (8)



119866 = 119878d119861119911

d119911(9)



119911




119881119871

(119905) = (119877119871

119877119871

+ 119877119862

)119866 (119905) (10)



119881119871

(120596) = (119877119871

119877119871

+ 119877119862

)119866 |119867 (119894120596)| 119860 (120596) (11)


1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816 = (

119877119871

119877119871

+ 119877119862

)119866 |119867 (119894120596)| (12)


119860


119878119881119871(120596) =

1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816

2

119878119860

(120596) (13)


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119878119860

(120596)

(14)


1198812

119871

= int

infin

minusinfin

119878119881119871(120596) d120596

= (119877119871

119877119871

+ 119877119862

)

2

1198662

int

infin

minusinfin

|119867 (119894120596)|2

119878119860

(120596) d120596(15)


119875119871

=1198812

119871

119877119871

=1

119877119871

int

infin

minusinfin

119878119881119871(120596) d120596 = int

infin

minusinfin

SPL(120596) d120596 (16)


119878119875119871(120596) =

1

119877119871

119878119881119871(120596) =

1

119877119871

(119877119871

119877119871

+ 119877119862

)

2

times 1198662

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119878119860

(120596)

(17)


119860


119860

(120596) = 1198780


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

int

infin

minusinfin

|119867 (119894120596)|2d120596 (18)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577119879

120596119899

(19)


119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577119879

120596119899

(20)


= 120577119898

+ 120577119890


120577119898

=1

2119876119898

(21)




120577119890

=1198662

2119898120596119899

(119877119871

+ 119877119862

)(22)




119875119871

= 1198981205871198780

119877119871

119877119871

+ 119877119862

1198662

119876119898

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

(23)



119877119871opt = radic119877

2

119862

+1198662

119876119898

119877119862

119898120596119899

= radic1198772

119862

+1198662

119877119862

119887119898

(24)


119871opt = 119877119862

+ 1198662

119887119898


The product 1198662119876119898


119898



119911


119911




2

119876119898




119903



119898


119860

(120596) = 1198780

=


2

119876119898


119898


119898


119898



119878119881119871(120596)

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

119899

)

2


120596119899

)

2

]

2

+ [(1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times(120596

120596119899

)]

2

)

minus1

)

(25)


Δ120596 = 2120577119879

120596119899

= 2 (120577119898

+ 120577119890

) 120596119899

=120596119899

119876119898

+1198662

119898(119877119871

+ 119877119862

) (26)



Table1Parameterso

fMEM

Sscalee

lectromagnetic

energy

harvesters

Coiltype

m(kg)

119865resonant(H

z)119877119862

(Ω)

119877119871

(Ω)

Mechqu

ality

factor119876119898

Electqu

ality

factor119876119890


G(Tm)

1198662

119876119898

(T2 m

2 )Re

ferences

Wou

nd

428times10minus3

1311

1811752

653

a3118

b114

252

[29]

25times10minus3

84365

3298

8a03821

b[29]

38times10minus3

948

1227

2587c

2461a

03322

b2855

[30]

044times10minus3

350

93100

216

1120

181

041

b3631

[31]

0028times10minus3

95k

100

164

[31]

521

100

200

232

243

119[6]

102times10minus3

502323

210ndash

60604

[32]

1530

443

[32]

0028times10minus3

808

k112

26[33]

208

1218

[34]

106

1413

[34]

Planar

4036times10minus3

248

100

100

2336

[19]

00304times10minus3

100

24

162

c40323

a79

4c0017b

0005

[35]

0014times10minus3

9837

k55

164

[33]

054times10minus3

60110

110485

c234533

a00691

b0232

[33]

3139

136

120

15times10minus3

000

03[27]

221d

207d

[27]

093times10minus3

371

75100

57

583

c0075

0033

[17]


n119876

119890

=12120577

119890

b C


n119866=radic(119877

119871

+119877

119862

)(2120587119898119865119876

119890

)


n119876

119898

=12120577

119898

=2120587119898119865119887

119898

d D

etermined

from

testing

invacuum




119862


119898



119898


119898





1198781198811198711

= 119878119881119871(120596 = 120596

119899

)

=1198772

119871

119877119871

+ 119877119862

1198981198780

1198662

119876119898

120596119899

(119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

)

(27)




119860

(120596) = 1198780


and

0

02

04

06

08

1

12

Mea

n po

wer

(W)


mG2Q = 1T2 m2

G2Qm = 10T2 m2

G2Qm = 100T2 m2

G2Qm = 500T2 m2

G2Qm = 750T2 m2

G2Qm = 1000T2 m2

times10minus4


119898

002040608

1121416

Mea

n po

wer

(W)


times10minus5

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2


119898

= 57

1205962


1

le |120596| le 1205962



= 1640 rads to 1205962



1198812

119871

= 1198780

[int

minus1205961

minus1205962

1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816

2d120596 + int1205962

1205961

1003816100381610038161003816HV (i120596)1003816100381610038161003816

2d120596] (28)


0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f lo

ad v

olta

ge (V

rms2

rad

s)


= 57

220240260280300320340360380

Band

wid

th (r

ads

)


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2


119898

= 57


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times [

[

int

minus1205961

minus1205962

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596

+int

1205962

1205961

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596]

]

(29)


0

2

4

6

022002300

240050 100 150 200

times10minus7


SD o

f pow

er (W

)


119898

= 57 and 119866 = 01Tm


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1205871198662

1198780

2120577119879

120596119899

[Γ(1205962

120596119899

120577119879

) minus Γ(1205961

120596119899

120577119879

)]

(30)


Γ(120596

120596119899

120577119879

) =1

120587arc tan

2120577119879

(120596120596119899

)

1 minus (120596120596119899

)2

minus120577119879

2120587radic1 minus 120577119879

2

times ln1 + (120596120596

119899

)2


119879

(120596120596119899

)

1 + (120596120596119899

)2


119879

(120596120596119899

)

(31)




0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


120577T = 002

120577T = 006

120577T = 01

Inte

gral

fact

orΓ


0

02

04

06

08

1

0 05 1 15 2



) minus

Γ(0 120577119879



= 120577119898

+

120577119890



119890





119898 + 119887119879

+ [119896119911 + 120578119896119873 (119911)] = minus119898 119910 (32)





(32) results in

+ 2120577119879

120596119899

+ 1205962

119899

(119911 + 1205781199113

) = minus 119910 (33)


119899

(119911 + 1205781199113


+ 2120577eq119879

120596eq + 1205962

eq119911 = minus 119910 (34)


=

(120596119899


] of the error

119890 = 1205962

119899

(119911 + 1205781199113

) minus 1205962

eq119911 (35)


dd1205962eq

119864 [1198902

] = 0 (36)



1205962

eq = 1205962

119899

119864 [119911119911 (1 + 1205781199112

)]

1205902

119911

(37)




1205962

eq = 1205962

119899

(1 + 31205781205902

119911

) (38)


1205902

119911

= int

infin

minusinfin

119878119911

(120596) 119889120596 = 1198780

int

infin

minusinfin

|119867 (119894120596)|2

119889120596 (39)


119867(119894120596) =minus1

minus1205962

+ 1198942120577eq119879

120596eq120596 + 1205962

eq(40)


1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

2120577119879

120596119899

1205962

eq (41)


119911

31205781205904

119911

+ 1205902

119911

= 1205902

119911Lin=

1205871198780

2120577119879

1205963

119899

(42)

where 120590119911Lin



1205902

119911

=

radic1 + 121205781205902

119911Linminus 1

6120578

(43)


1205962

eq = 1205962

119899

[[

[

1 +

radic1 + 121205781205902

119911Linminus 1

2

]]

]

(44)


1

119866(119877119871

+ 119877119862

119877119871

)[ddt119881119871

(119905) + 2120577eq119879

120596eq119881119871 (119905) + 1205962

eq int119881119871 (119905) dt]

= minus 119910 (119905)

(45)


119867119881

(119894120596) = (119877119871

119877119871

+ 119877119862

)119866120596

(1205962

eq minus 1205962

) 119894 minus 2120577eq119879

120596eq120596 (46)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

1198780

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

eq)

2


120596eq)

2

]

2

+ [(120596119899

120596eq)(

1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times (120596

120596eq)]

2

)

minus1

)

(47)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(48)


119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(49)



119903



119898





0

05

1

15

2

25

3




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)





+ 119887119879

120572119863()


119898 + [119887119879

+ 119887119879

120572119863 ()] + [119896119911 + 119896120578119873 (119911)] = minus119898 119910 (50)

0

1

2

3

4


times10minus4



SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)



+ 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

) = minus 119910 (51)


+ 119887119879

1205723



119879




120596119899

( + 1205723


(119911 + 1205781199113

)



+ 120583eq + 1205962

eq119911 = minus 119910 (52)


119890 = 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113


eq119911 (53)


120597

1205971205962

eq119864 [1198902

] = 0

120597

120597120583eq119864 [1198902

] = 0

(54)



119864 [2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [2

] minus 1205962

eq119864 [119911] = 0

119864 [1199112120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [119911] minus 1205962

eq119864 [1199112

] = 0

(55)



2

119911

and 119864[2] = 1205902

119911


120583eq =119864 [2120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(56)



1205962

eq =119864 [1199112120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(57)



120583eq = 119864[120597

120597[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

1205962

eq = 119864[120597

120597119911[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

(58)


120583eq = 119864 [2120577119879120596119899 (1 + 31205722

)] = 120583119879

(1 + 31205721205902

119911

) (59)


= 2120577119879

120596119899


1205962

eq = 119864 [1205962

119899

(1 + 31205781199112

)] = 1205962

119899

(1 + 31205781205902

119911

) (60)




1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

120583eq1205962

eq(61)


1205902

119911

=1205871198780

2120577eq119879

120596eq=1205871198780

120583eq(62)



1205902

119911119871

=1205871198780

2120577119879120596119899

=1205871198780

120583119879

(63)



119911

1205902

119911

(1 + 31205721205902

119911

) = 1205902

119911119871

(64)


1205902

119911

1205902

119911119871

=120583119879

120583eq=

radic1 + 121205721205902

119911119871

minus 1

61205721205902

119911119871

(65)


120583eq = 120583119879[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

(66)


1205902

119911119871

=1205871198780

2120577119879

1205963

119899

=1205871198780

120583119879

1205962

119899

(67)



119911

1205902

119911

(1 + 31205781205902

119911

)

120583eq

120583119879

= 1205902

119911119871

(68)


1205902

119911

1205902

119911119871

= (radic1 + 12119871 minus 1

6119871)120583119879

120583eq(69)


1205962

eq = 1205962

119899

[1 +radic1 + 12119871 minus 1

2] (70)

where

119871 = 1205781205902

119911119871

120583119879

120583eq= 1205781205902

119911119871

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

minus1

(71)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

(72)





120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)


119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)




0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


005 s2mminus2







002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120577119890

=1198662

2119898120596119899

(119877119871

+ 119877119862

)(22)




119875119871

= 1198981205871198780

119877119871

119877119871

+ 119877119862

1198662

119876119898

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

(23)



119877119871opt = radic119877

2

119862

+1198662

119876119898

119877119862

119898120596119899

= radic1198772

119862

+1198662

119877119862

119887119898

(24)


119871opt = 119877119862

+ 1198662

119887119898


The product 1198662119876119898


119898



119911


119911




2

119876119898




119903



119898


119860

(120596) = 1198780

=


2

119876119898


119898


119898


119898



119878119881119871(120596)

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

119899

)

2


120596119899

)

2

]

2

+ [(1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times(120596

120596119899

)]

2

)

minus1

)

(25)


Δ120596 = 2120577119879

120596119899

= 2 (120577119898

+ 120577119890

) 120596119899

=120596119899

119876119898

+1198662

119898(119877119871

+ 119877119862

) (26)



Table1Parameterso

fMEM

Sscalee

lectromagnetic

energy

harvesters

Coiltype

m(kg)

119865resonant(H

z)119877119862

(Ω)

119877119871

(Ω)

Mechqu

ality

factor119876119898

Electqu

ality

factor119876119890


G(Tm)

1198662

119876119898

(T2 m

2 )Re

ferences

Wou

nd

428times10minus3

1311

1811752

653

a3118

b114

252

[29]

25times10minus3

84365

3298

8a03821

b[29]

38times10minus3

948

1227

2587c

2461a

03322

b2855

[30]

044times10minus3

350

93100

216

1120

181

041

b3631

[31]

0028times10minus3

95k

100

164

[31]

521

100

200

232

243

119[6]

102times10minus3

502323

210ndash

60604

[32]

1530

443

[32]

0028times10minus3

808

k112

26[33]

208

1218

[34]

106

1413

[34]

Planar

4036times10minus3

248

100

100

2336

[19]

00304times10minus3

100

24

162

c40323

a79

4c0017b

0005

[35]

0014times10minus3

9837

k55

164

[33]

054times10minus3

60110

110485

c234533

a00691

b0232

[33]

3139

136

120

15times10minus3

000

03[27]

221d

207d

[27]

093times10minus3

371

75100

57

583

c0075

0033

[17]


n119876

119890

=12120577

119890

b C


n119866=radic(119877

119871

+119877

119862

)(2120587119898119865119876

119890

)


n119876

119898

=12120577

119898

=2120587119898119865119887

119898

d D

etermined

from

testing

invacuum




119862


119898



119898


119898





1198781198811198711

= 119878119881119871(120596 = 120596

119899

)

=1198772

119871

119877119871

+ 119877119862

1198981198780

1198662

119876119898

120596119899

(119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

)

(27)




119860

(120596) = 1198780


and

0

02

04

06

08

1

12

Mea

n po

wer

(W)


mG2Q = 1T2 m2

G2Qm = 10T2 m2

G2Qm = 100T2 m2

G2Qm = 500T2 m2

G2Qm = 750T2 m2

G2Qm = 1000T2 m2

times10minus4


119898

002040608

1121416

Mea

n po

wer

(W)


times10minus5

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2


119898

= 57

1205962


1

le |120596| le 1205962



= 1640 rads to 1205962



1198812

119871

= 1198780

[int

minus1205961

minus1205962

1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816

2d120596 + int1205962

1205961

1003816100381610038161003816HV (i120596)1003816100381610038161003816

2d120596] (28)


0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f lo

ad v

olta

ge (V

rms2

rad

s)


= 57

220240260280300320340360380

Band

wid

th (r

ads

)


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2


119898

= 57


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times [

[

int

minus1205961

minus1205962

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596

+int

1205962

1205961

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596]

]

(29)


0

2

4

6

022002300

240050 100 150 200

times10minus7


SD o

f pow

er (W

)


119898

= 57 and 119866 = 01Tm


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1205871198662

1198780

2120577119879

120596119899

[Γ(1205962

120596119899

120577119879

) minus Γ(1205961

120596119899

120577119879

)]

(30)


Γ(120596

120596119899

120577119879

) =1

120587arc tan

2120577119879

(120596120596119899

)

1 minus (120596120596119899

)2

minus120577119879

2120587radic1 minus 120577119879

2

times ln1 + (120596120596

119899

)2


119879

(120596120596119899

)

1 + (120596120596119899

)2


119879

(120596120596119899

)

(31)




0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


120577T = 002

120577T = 006

120577T = 01

Inte

gral

fact

orΓ


0

02

04

06

08

1

0 05 1 15 2



) minus

Γ(0 120577119879



= 120577119898

+

120577119890



119890





119898 + 119887119879

+ [119896119911 + 120578119896119873 (119911)] = minus119898 119910 (32)





(32) results in

+ 2120577119879

120596119899

+ 1205962

119899

(119911 + 1205781199113

) = minus 119910 (33)


119899

(119911 + 1205781199113


+ 2120577eq119879

120596eq + 1205962

eq119911 = minus 119910 (34)


=

(120596119899


] of the error

119890 = 1205962

119899

(119911 + 1205781199113

) minus 1205962

eq119911 (35)


dd1205962eq

119864 [1198902

] = 0 (36)



1205962

eq = 1205962

119899

119864 [119911119911 (1 + 1205781199112

)]

1205902

119911

(37)




1205962

eq = 1205962

119899

(1 + 31205781205902

119911

) (38)


1205902

119911

= int

infin

minusinfin

119878119911

(120596) 119889120596 = 1198780

int

infin

minusinfin

|119867 (119894120596)|2

119889120596 (39)


119867(119894120596) =minus1

minus1205962

+ 1198942120577eq119879

120596eq120596 + 1205962

eq(40)


1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

2120577119879

120596119899

1205962

eq (41)


119911

31205781205904

119911

+ 1205902

119911

= 1205902

119911Lin=

1205871198780

2120577119879

1205963

119899

(42)

where 120590119911Lin



1205902

119911

=

radic1 + 121205781205902

119911Linminus 1

6120578

(43)


1205962

eq = 1205962

119899

[[

[

1 +

radic1 + 121205781205902

119911Linminus 1

2

]]

]

(44)


1

119866(119877119871

+ 119877119862

119877119871

)[ddt119881119871

(119905) + 2120577eq119879

120596eq119881119871 (119905) + 1205962

eq int119881119871 (119905) dt]

= minus 119910 (119905)

(45)


119867119881

(119894120596) = (119877119871

119877119871

+ 119877119862

)119866120596

(1205962

eq minus 1205962

) 119894 minus 2120577eq119879

120596eq120596 (46)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

1198780

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

eq)

2


120596eq)

2

]

2

+ [(120596119899

120596eq)(

1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times (120596

120596eq)]

2

)

minus1

)

(47)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(48)


119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(49)



119903



119898





0

05

1

15

2

25

3




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)





+ 119887119879

120572119863()


119898 + [119887119879

+ 119887119879

120572119863 ()] + [119896119911 + 119896120578119873 (119911)] = minus119898 119910 (50)

0

1

2

3

4


times10minus4



SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)



+ 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

) = minus 119910 (51)


+ 119887119879

1205723



119879




120596119899

( + 1205723


(119911 + 1205781199113

)



+ 120583eq + 1205962

eq119911 = minus 119910 (52)


119890 = 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113


eq119911 (53)


120597

1205971205962

eq119864 [1198902

] = 0

120597

120597120583eq119864 [1198902

] = 0

(54)



119864 [2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [2

] minus 1205962

eq119864 [119911] = 0

119864 [1199112120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [119911] minus 1205962

eq119864 [1199112

] = 0

(55)



2

119911

and 119864[2] = 1205902

119911


120583eq =119864 [2120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(56)



1205962

eq =119864 [1199112120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(57)



120583eq = 119864[120597

120597[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

1205962

eq = 119864[120597

120597119911[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

(58)


120583eq = 119864 [2120577119879120596119899 (1 + 31205722

)] = 120583119879

(1 + 31205721205902

119911

) (59)


= 2120577119879

120596119899


1205962

eq = 119864 [1205962

119899

(1 + 31205781199112

)] = 1205962

119899

(1 + 31205781205902

119911

) (60)




1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

120583eq1205962

eq(61)


1205902

119911

=1205871198780

2120577eq119879

120596eq=1205871198780

120583eq(62)



1205902

119911119871

=1205871198780

2120577119879120596119899

=1205871198780

120583119879

(63)



119911

1205902

119911

(1 + 31205721205902

119911

) = 1205902

119911119871

(64)


1205902

119911

1205902

119911119871

=120583119879

120583eq=

radic1 + 121205721205902

119911119871

minus 1

61205721205902

119911119871

(65)


120583eq = 120583119879[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

(66)


1205902

119911119871

=1205871198780

2120577119879

1205963

119899

=1205871198780

120583119879

1205962

119899

(67)



119911

1205902

119911

(1 + 31205781205902

119911

)

120583eq

120583119879

= 1205902

119911119871

(68)


1205902

119911

1205902

119911119871

= (radic1 + 12119871 minus 1

6119871)120583119879

120583eq(69)


1205962

eq = 1205962

119899

[1 +radic1 + 12119871 minus 1

2] (70)

where

119871 = 1205781205902

119911119871

120583119879

120583eq= 1205781205902

119911119871

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

minus1

(71)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

(72)





120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)


119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)




0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


005 s2mminus2







002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table1Parameterso

fMEM

Sscalee

lectromagnetic

energy

harvesters

Coiltype

m(kg)

119865resonant(H

z)119877119862

(Ω)

119877119871

(Ω)

Mechqu

ality

factor119876119898

Electqu

ality

factor119876119890


G(Tm)

1198662

119876119898

(T2 m

2 )Re

ferences

Wou

nd

428times10minus3

1311

1811752

653

a3118

b114

252

[29]

25times10minus3

84365

3298

8a03821

b[29]

38times10minus3

948

1227

2587c

2461a

03322

b2855

[30]

044times10minus3

350

93100

216

1120

181

041

b3631

[31]

0028times10minus3

95k

100

164

[31]

521

100

200

232

243

119[6]

102times10minus3

502323

210ndash

60604

[32]

1530

443

[32]

0028times10minus3

808

k112

26[33]

208

1218

[34]

106

1413

[34]

Planar

4036times10minus3

248

100

100

2336

[19]

00304times10minus3

100

24

162

c40323

a79

4c0017b

0005

[35]

0014times10minus3

9837

k55

164

[33]

054times10minus3

60110

110485

c234533

a00691

b0232

[33]

3139

136

120

15times10minus3

000

03[27]

221d

207d

[27]

093times10minus3

371

75100

57

583

c0075

0033

[17]


n119876

119890

=12120577

119890

b C


n119866=radic(119877

119871

+119877

119862

)(2120587119898119865119876

119890

)


n119876

119898

=12120577

119898

=2120587119898119865119887

119898

d D

etermined

from

testing

invacuum




119862


119898



119898


119898





1198781198811198711

= 119878119881119871(120596 = 120596

119899

)

=1198772

119871

119877119871

+ 119877119862

1198981198780

1198662

119876119898

120596119899

(119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

)

(27)




119860

(120596) = 1198780


and

0

02

04

06

08

1

12

Mea

n po

wer

(W)


mG2Q = 1T2 m2

G2Qm = 10T2 m2

G2Qm = 100T2 m2

G2Qm = 500T2 m2

G2Qm = 750T2 m2

G2Qm = 1000T2 m2

times10minus4


119898

002040608

1121416

Mea

n po

wer

(W)


times10minus5

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2


119898

= 57

1205962


1

le |120596| le 1205962



= 1640 rads to 1205962



1198812

119871

= 1198780

[int

minus1205961

minus1205962

1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816

2d120596 + int1205962

1205961

1003816100381610038161003816HV (i120596)1003816100381610038161003816

2d120596] (28)


0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f lo

ad v

olta

ge (V

rms2

rad

s)


= 57

220240260280300320340360380

Band

wid

th (r

ads

)


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2


119898

= 57


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times [

[

int

minus1205961

minus1205962

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596

+int

1205962

1205961

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596]

]

(29)


0

2

4

6

022002300

240050 100 150 200

times10minus7


SD o

f pow

er (W

)


119898

= 57 and 119866 = 01Tm


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1205871198662

1198780

2120577119879

120596119899

[Γ(1205962

120596119899

120577119879

) minus Γ(1205961

120596119899

120577119879

)]

(30)


Γ(120596

120596119899

120577119879

) =1

120587arc tan

2120577119879

(120596120596119899

)

1 minus (120596120596119899

)2

minus120577119879

2120587radic1 minus 120577119879

2

times ln1 + (120596120596

119899

)2


119879

(120596120596119899

)

1 + (120596120596119899

)2


119879

(120596120596119899

)

(31)




0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


120577T = 002

120577T = 006

120577T = 01

Inte

gral

fact

orΓ


0

02

04

06

08

1

0 05 1 15 2



) minus

Γ(0 120577119879



= 120577119898

+

120577119890



119890





119898 + 119887119879

+ [119896119911 + 120578119896119873 (119911)] = minus119898 119910 (32)





(32) results in

+ 2120577119879

120596119899

+ 1205962

119899

(119911 + 1205781199113

) = minus 119910 (33)


119899

(119911 + 1205781199113


+ 2120577eq119879

120596eq + 1205962

eq119911 = minus 119910 (34)


=

(120596119899


] of the error

119890 = 1205962

119899

(119911 + 1205781199113

) minus 1205962

eq119911 (35)


dd1205962eq

119864 [1198902

] = 0 (36)



1205962

eq = 1205962

119899

119864 [119911119911 (1 + 1205781199112

)]

1205902

119911

(37)




1205962

eq = 1205962

119899

(1 + 31205781205902

119911

) (38)


1205902

119911

= int

infin

minusinfin

119878119911

(120596) 119889120596 = 1198780

int

infin

minusinfin

|119867 (119894120596)|2

119889120596 (39)


119867(119894120596) =minus1

minus1205962

+ 1198942120577eq119879

120596eq120596 + 1205962

eq(40)


1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

2120577119879

120596119899

1205962

eq (41)


119911

31205781205904

119911

+ 1205902

119911

= 1205902

119911Lin=

1205871198780

2120577119879

1205963

119899

(42)

where 120590119911Lin



1205902

119911

=

radic1 + 121205781205902

119911Linminus 1

6120578

(43)


1205962

eq = 1205962

119899

[[

[

1 +

radic1 + 121205781205902

119911Linminus 1

2

]]

]

(44)


1

119866(119877119871

+ 119877119862

119877119871

)[ddt119881119871

(119905) + 2120577eq119879

120596eq119881119871 (119905) + 1205962

eq int119881119871 (119905) dt]

= minus 119910 (119905)

(45)


119867119881

(119894120596) = (119877119871

119877119871

+ 119877119862

)119866120596

(1205962

eq minus 1205962

) 119894 minus 2120577eq119879

120596eq120596 (46)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

1198780

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

eq)

2


120596eq)

2

]

2

+ [(120596119899

120596eq)(

1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times (120596

120596eq)]

2

)

minus1

)

(47)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(48)


119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(49)



119903



119898





0

05

1

15

2

25

3




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)





+ 119887119879

120572119863()


119898 + [119887119879

+ 119887119879

120572119863 ()] + [119896119911 + 119896120578119873 (119911)] = minus119898 119910 (50)

0

1

2

3

4


times10minus4



SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)



+ 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

) = minus 119910 (51)


+ 119887119879

1205723



119879




120596119899

( + 1205723


(119911 + 1205781199113

)



+ 120583eq + 1205962

eq119911 = minus 119910 (52)


119890 = 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113


eq119911 (53)


120597

1205971205962

eq119864 [1198902

] = 0

120597

120597120583eq119864 [1198902

] = 0

(54)



119864 [2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [2

] minus 1205962

eq119864 [119911] = 0

119864 [1199112120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [119911] minus 1205962

eq119864 [1199112

] = 0

(55)



2

119911

and 119864[2] = 1205902

119911


120583eq =119864 [2120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(56)



1205962

eq =119864 [1199112120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(57)



120583eq = 119864[120597

120597[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

1205962

eq = 119864[120597

120597119911[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

(58)


120583eq = 119864 [2120577119879120596119899 (1 + 31205722

)] = 120583119879

(1 + 31205721205902

119911

) (59)


= 2120577119879

120596119899


1205962

eq = 119864 [1205962

119899

(1 + 31205781199112

)] = 1205962

119899

(1 + 31205781205902

119911

) (60)




1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

120583eq1205962

eq(61)


1205902

119911

=1205871198780

2120577eq119879

120596eq=1205871198780

120583eq(62)



1205902

119911119871

=1205871198780

2120577119879120596119899

=1205871198780

120583119879

(63)



119911

1205902

119911

(1 + 31205721205902

119911

) = 1205902

119911119871

(64)


1205902

119911

1205902

119911119871

=120583119879

120583eq=

radic1 + 121205721205902

119911119871

minus 1

61205721205902

119911119871

(65)


120583eq = 120583119879[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

(66)


1205902

119911119871

=1205871198780

2120577119879

1205963

119899

=1205871198780

120583119879

1205962

119899

(67)



119911

1205902

119911

(1 + 31205781205902

119911

)

120583eq

120583119879

= 1205902

119911119871

(68)


1205902

119911

1205902

119911119871

= (radic1 + 12119871 minus 1

6119871)120583119879

120583eq(69)


1205962

eq = 1205962

119899

[1 +radic1 + 12119871 minus 1

2] (70)

where

119871 = 1205781205902

119911119871

120583119879

120583eq= 1205781205902

119911119871

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

minus1

(71)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

(72)





120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)


119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)




0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


005 s2mminus2







002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119862


119898



119898


119898





1198781198811198711

= 119878119881119871(120596 = 120596

119899

)

=1198772

119871

119877119871

+ 119877119862

1198981198780

1198662

119876119898

120596119899

(119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

)

(27)




119860

(120596) = 1198780


and

0

02

04

06

08

1

12

Mea

n po

wer

(W)


mG2Q = 1T2 m2

G2Qm = 10T2 m2

G2Qm = 100T2 m2

G2Qm = 500T2 m2

G2Qm = 750T2 m2

G2Qm = 1000T2 m2

times10minus4


119898

002040608

1121416

Mea

n po

wer

(W)


times10minus5

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2


119898

= 57

1205962


1

le |120596| le 1205962



= 1640 rads to 1205962



1198812

119871

= 1198780

[int

minus1205961

minus1205962

1003816100381610038161003816119867119881 (119894120596)1003816100381610038161003816

2d120596 + int1205962

1205961

1003816100381610038161003816HV (i120596)1003816100381610038161003816

2d120596] (28)


0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f lo

ad v

olta

ge (V

rms2

rad

s)


= 57

220240260280300320340360380

Band

wid

th (r

ads

)


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2


119898

= 57


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times [

[

int

minus1205961

minus1205962

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596

+int

1205962

1205961

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596]

]

(29)


0

2

4

6

022002300

240050 100 150 200

times10minus7


SD o

f pow

er (W

)


119898

= 57 and 119866 = 01Tm


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1205871198662

1198780

2120577119879

120596119899

[Γ(1205962

120596119899

120577119879

) minus Γ(1205961

120596119899

120577119879

)]

(30)


Γ(120596

120596119899

120577119879

) =1

120587arc tan

2120577119879

(120596120596119899

)

1 minus (120596120596119899

)2

minus120577119879

2120587radic1 minus 120577119879

2

times ln1 + (120596120596

119899

)2


119879

(120596120596119899

)

1 + (120596120596119899

)2


119879

(120596120596119899

)

(31)




0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


120577T = 002

120577T = 006

120577T = 01

Inte

gral

fact

orΓ


0

02

04

06

08

1

0 05 1 15 2



) minus

Γ(0 120577119879



= 120577119898

+

120577119890



119890





119898 + 119887119879

+ [119896119911 + 120578119896119873 (119911)] = minus119898 119910 (32)





(32) results in

+ 2120577119879

120596119899

+ 1205962

119899

(119911 + 1205781199113

) = minus 119910 (33)


119899

(119911 + 1205781199113


+ 2120577eq119879

120596eq + 1205962

eq119911 = minus 119910 (34)


=

(120596119899


] of the error

119890 = 1205962

119899

(119911 + 1205781199113

) minus 1205962

eq119911 (35)


dd1205962eq

119864 [1198902

] = 0 (36)



1205962

eq = 1205962

119899

119864 [119911119911 (1 + 1205781199112

)]

1205902

119911

(37)




1205962

eq = 1205962

119899

(1 + 31205781205902

119911

) (38)


1205902

119911

= int

infin

minusinfin

119878119911

(120596) 119889120596 = 1198780

int

infin

minusinfin

|119867 (119894120596)|2

119889120596 (39)


119867(119894120596) =minus1

minus1205962

+ 1198942120577eq119879

120596eq120596 + 1205962

eq(40)


1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

2120577119879

120596119899

1205962

eq (41)


119911

31205781205904

119911

+ 1205902

119911

= 1205902

119911Lin=

1205871198780

2120577119879

1205963

119899

(42)

where 120590119911Lin



1205902

119911

=

radic1 + 121205781205902

119911Linminus 1

6120578

(43)


1205962

eq = 1205962

119899

[[

[

1 +

radic1 + 121205781205902

119911Linminus 1

2

]]

]

(44)


1

119866(119877119871

+ 119877119862

119877119871

)[ddt119881119871

(119905) + 2120577eq119879

120596eq119881119871 (119905) + 1205962

eq int119881119871 (119905) dt]

= minus 119910 (119905)

(45)


119867119881

(119894120596) = (119877119871

119877119871

+ 119877119862

)119866120596

(1205962

eq minus 1205962

) 119894 minus 2120577eq119879

120596eq120596 (46)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

1198780

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

eq)

2


120596eq)

2

]

2

+ [(120596119899

120596eq)(

1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times (120596

120596eq)]

2

)

minus1

)

(47)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(48)


119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(49)



119903



119898





0

05

1

15

2

25

3




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)





+ 119887119879

120572119863()


119898 + [119887119879

+ 119887119879

120572119863 ()] + [119896119911 + 119896120578119873 (119911)] = minus119898 119910 (50)

0

1

2

3

4


times10minus4



SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)



+ 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

) = minus 119910 (51)


+ 119887119879

1205723



119879




120596119899

( + 1205723


(119911 + 1205781199113

)



+ 120583eq + 1205962

eq119911 = minus 119910 (52)


119890 = 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113


eq119911 (53)


120597

1205971205962

eq119864 [1198902

] = 0

120597

120597120583eq119864 [1198902

] = 0

(54)



119864 [2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [2

] minus 1205962

eq119864 [119911] = 0

119864 [1199112120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [119911] minus 1205962

eq119864 [1199112

] = 0

(55)



2

119911

and 119864[2] = 1205902

119911


120583eq =119864 [2120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(56)



1205962

eq =119864 [1199112120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(57)



120583eq = 119864[120597

120597[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

1205962

eq = 119864[120597

120597119911[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

(58)


120583eq = 119864 [2120577119879120596119899 (1 + 31205722

)] = 120583119879

(1 + 31205721205902

119911

) (59)


= 2120577119879

120596119899


1205962

eq = 119864 [1205962

119899

(1 + 31205781199112

)] = 1205962

119899

(1 + 31205781205902

119911

) (60)




1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

120583eq1205962

eq(61)


1205902

119911

=1205871198780

2120577eq119879

120596eq=1205871198780

120583eq(62)



1205902

119911119871

=1205871198780

2120577119879120596119899

=1205871198780

120583119879

(63)



119911

1205902

119911

(1 + 31205721205902

119911

) = 1205902

119911119871

(64)


1205902

119911

1205902

119911119871

=120583119879

120583eq=

radic1 + 121205721205902

119911119871

minus 1

61205721205902

119911119871

(65)


120583eq = 120583119879[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

(66)


1205902

119911119871

=1205871198780

2120577119879

1205963

119899

=1205871198780

120583119879

1205962

119899

(67)



119911

1205902

119911

(1 + 31205781205902

119911

)

120583eq

120583119879

= 1205902

119911119871

(68)


1205902

119911

1205902

119911119871

= (radic1 + 12119871 minus 1

6119871)120583119879

120583eq(69)


1205962

eq = 1205962

119899

[1 +radic1 + 12119871 minus 1

2] (70)

where

119871 = 1205781205902

119911119871

120583119879

120583eq= 1205781205902

119911119871

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

minus1

(71)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

(72)





120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)


119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)




0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


005 s2mminus2







002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f lo

ad v

olta

ge (V

rms2

rad

s)


= 57

220240260280300320340360380

Band

wid

th (r

ads

)


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2


119898

= 57


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times [

[

int

minus1205961

minus1205962

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596

+int

1205962

1205961

(1205961205962

119899

)2

[1 minus (120596120596119899

)2

]2

+ [2120577119879

(120596120596119899

)]2

119889120596]

]

(29)


0

2

4

6

022002300

240050 100 150 200

times10minus7


SD o

f pow

er (W

)


119898

= 57 and 119866 = 01Tm


1198812

119871

= (119877119871

119877119871

+ 119877119862

)

2

1205871198662

1198780

2120577119879

120596119899

[Γ(1205962

120596119899

120577119879

) minus Γ(1205961

120596119899

120577119879

)]

(30)


Γ(120596

120596119899

120577119879

) =1

120587arc tan

2120577119879

(120596120596119899

)

1 minus (120596120596119899

)2

minus120577119879

2120587radic1 minus 120577119879

2

times ln1 + (120596120596

119899

)2


119879

(120596120596119899

)

1 + (120596120596119899

)2


119879

(120596120596119899

)

(31)




0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


120577T = 002

120577T = 006

120577T = 01

Inte

gral

fact

orΓ


0

02

04

06

08

1

0 05 1 15 2



) minus

Γ(0 120577119879



= 120577119898

+

120577119890



119890





119898 + 119887119879

+ [119896119911 + 120578119896119873 (119911)] = minus119898 119910 (32)





(32) results in

+ 2120577119879

120596119899

+ 1205962

119899

(119911 + 1205781199113

) = minus 119910 (33)


119899

(119911 + 1205781199113


+ 2120577eq119879

120596eq + 1205962

eq119911 = minus 119910 (34)


=

(120596119899


] of the error

119890 = 1205962

119899

(119911 + 1205781199113

) minus 1205962

eq119911 (35)


dd1205962eq

119864 [1198902

] = 0 (36)



1205962

eq = 1205962

119899

119864 [119911119911 (1 + 1205781199112

)]

1205902

119911

(37)




1205962

eq = 1205962

119899

(1 + 31205781205902

119911

) (38)


1205902

119911

= int

infin

minusinfin

119878119911

(120596) 119889120596 = 1198780

int

infin

minusinfin

|119867 (119894120596)|2

119889120596 (39)


119867(119894120596) =minus1

minus1205962

+ 1198942120577eq119879

120596eq120596 + 1205962

eq(40)


1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

2120577119879

120596119899

1205962

eq (41)


119911

31205781205904

119911

+ 1205902

119911

= 1205902

119911Lin=

1205871198780

2120577119879

1205963

119899

(42)

where 120590119911Lin



1205902

119911

=

radic1 + 121205781205902

119911Linminus 1

6120578

(43)


1205962

eq = 1205962

119899

[[

[

1 +

radic1 + 121205781205902

119911Linminus 1

2

]]

]

(44)


1

119866(119877119871

+ 119877119862

119877119871

)[ddt119881119871

(119905) + 2120577eq119879

120596eq119881119871 (119905) + 1205962

eq int119881119871 (119905) dt]

= minus 119910 (119905)

(45)


119867119881

(119894120596) = (119877119871

119877119871

+ 119877119862

)119866120596

(1205962

eq minus 1205962

) 119894 minus 2120577eq119879

120596eq120596 (46)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

1198780

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

eq)

2


120596eq)

2

]

2

+ [(120596119899

120596eq)(

1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times (120596

120596eq)]

2

)

minus1

)

(47)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(48)


119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(49)



119903



119898





0

05

1

15

2

25

3




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)





+ 119887119879

120572119863()


119898 + [119887119879

+ 119887119879

120572119863 ()] + [119896119911 + 119896120578119873 (119911)] = minus119898 119910 (50)

0

1

2

3

4


times10minus4



SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)



+ 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

) = minus 119910 (51)


+ 119887119879

1205723



119879




120596119899

( + 1205723


(119911 + 1205781199113

)



+ 120583eq + 1205962

eq119911 = minus 119910 (52)


119890 = 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113


eq119911 (53)


120597

1205971205962

eq119864 [1198902

] = 0

120597

120597120583eq119864 [1198902

] = 0

(54)



119864 [2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [2

] minus 1205962

eq119864 [119911] = 0

119864 [1199112120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [119911] minus 1205962

eq119864 [1199112

] = 0

(55)



2

119911

and 119864[2] = 1205902

119911


120583eq =119864 [2120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(56)



1205962

eq =119864 [1199112120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(57)



120583eq = 119864[120597

120597[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

1205962

eq = 119864[120597

120597119911[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

(58)


120583eq = 119864 [2120577119879120596119899 (1 + 31205722

)] = 120583119879

(1 + 31205721205902

119911

) (59)


= 2120577119879

120596119899


1205962

eq = 119864 [1205962

119899

(1 + 31205781199112

)] = 1205962

119899

(1 + 31205781205902

119911

) (60)




1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

120583eq1205962

eq(61)


1205902

119911

=1205871198780

2120577eq119879

120596eq=1205871198780

120583eq(62)



1205902

119911119871

=1205871198780

2120577119879120596119899

=1205871198780

120583119879

(63)



119911

1205902

119911

(1 + 31205721205902

119911

) = 1205902

119911119871

(64)


1205902

119911

1205902

119911119871

=120583119879

120583eq=

radic1 + 121205721205902

119911119871

minus 1

61205721205902

119911119871

(65)


120583eq = 120583119879[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

(66)


1205902

119911119871

=1205871198780

2120577119879

1205963

119899

=1205871198780

120583119879

1205962

119899

(67)



119911

1205902

119911

(1 + 31205781205902

119911

)

120583eq

120583119879

= 1205902

119911119871

(68)


1205902

119911

1205902

119911119871

= (radic1 + 12119871 minus 1

6119871)120583119879

120583eq(69)


1205962

eq = 1205962

119899

[1 +radic1 + 12119871 minus 1

2] (70)

where

119871 = 1205781205902

119911119871

120583119879

120583eq= 1205781205902

119911119871

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

minus1

(71)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

(72)





120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)


119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)




0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


005 s2mminus2







002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

05

1

15


G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2

times10minus5

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


120577T = 002

120577T = 006

120577T = 01

Inte

gral

fact

orΓ


0

02

04

06

08

1

0 05 1 15 2



) minus

Γ(0 120577119879



= 120577119898

+

120577119890



119890





119898 + 119887119879

+ [119896119911 + 120578119896119873 (119911)] = minus119898 119910 (32)





(32) results in

+ 2120577119879

120596119899

+ 1205962

119899

(119911 + 1205781199113

) = minus 119910 (33)


119899

(119911 + 1205781199113


+ 2120577eq119879

120596eq + 1205962

eq119911 = minus 119910 (34)


=

(120596119899


] of the error

119890 = 1205962

119899

(119911 + 1205781199113

) minus 1205962

eq119911 (35)


dd1205962eq

119864 [1198902

] = 0 (36)



1205962

eq = 1205962

119899

119864 [119911119911 (1 + 1205781199112

)]

1205902

119911

(37)




1205962

eq = 1205962

119899

(1 + 31205781205902

119911

) (38)


1205902

119911

= int

infin

minusinfin

119878119911

(120596) 119889120596 = 1198780

int

infin

minusinfin

|119867 (119894120596)|2

119889120596 (39)


119867(119894120596) =minus1

minus1205962

+ 1198942120577eq119879

120596eq120596 + 1205962

eq(40)


1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

2120577119879

120596119899

1205962

eq (41)


119911

31205781205904

119911

+ 1205902

119911

= 1205902

119911Lin=

1205871198780

2120577119879

1205963

119899

(42)

where 120590119911Lin



1205902

119911

=

radic1 + 121205781205902

119911Linminus 1

6120578

(43)


1205962

eq = 1205962

119899

[[

[

1 +

radic1 + 121205781205902

119911Linminus 1

2

]]

]

(44)


1

119866(119877119871

+ 119877119862

119877119871

)[ddt119881119871

(119905) + 2120577eq119879

120596eq119881119871 (119905) + 1205962

eq int119881119871 (119905) dt]

= minus 119910 (119905)

(45)


119867119881

(119894120596) = (119877119871

119877119871

+ 119877119862

)119866120596

(1205962

eq minus 1205962

) 119894 minus 2120577eq119879

120596eq120596 (46)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

1198780

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

eq)

2


120596eq)

2

]

2

+ [(120596119899

120596eq)(

1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times (120596

120596eq)]

2

)

minus1

)

(47)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(48)


119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(49)



119903



119898





0

05

1

15

2

25

3




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)





+ 119887119879

120572119863()


119898 + [119887119879

+ 119887119879

120572119863 ()] + [119896119911 + 119896120578119873 (119911)] = minus119898 119910 (50)

0

1

2

3

4


times10minus4



SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)



+ 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

) = minus 119910 (51)


+ 119887119879

1205723



119879




120596119899

( + 1205723


(119911 + 1205781199113

)



+ 120583eq + 1205962

eq119911 = minus 119910 (52)


119890 = 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113


eq119911 (53)


120597

1205971205962

eq119864 [1198902

] = 0

120597

120597120583eq119864 [1198902

] = 0

(54)



119864 [2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [2

] minus 1205962

eq119864 [119911] = 0

119864 [1199112120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [119911] minus 1205962

eq119864 [1199112

] = 0

(55)



2

119911

and 119864[2] = 1205902

119911


120583eq =119864 [2120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(56)



1205962

eq =119864 [1199112120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(57)



120583eq = 119864[120597

120597[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

1205962

eq = 119864[120597

120597119911[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

(58)


120583eq = 119864 [2120577119879120596119899 (1 + 31205722

)] = 120583119879

(1 + 31205721205902

119911

) (59)


= 2120577119879

120596119899


1205962

eq = 119864 [1205962

119899

(1 + 31205781199112

)] = 1205962

119899

(1 + 31205781205902

119911

) (60)




1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

120583eq1205962

eq(61)


1205902

119911

=1205871198780

2120577eq119879

120596eq=1205871198780

120583eq(62)



1205902

119911119871

=1205871198780

2120577119879120596119899

=1205871198780

120583119879

(63)



119911

1205902

119911

(1 + 31205721205902

119911

) = 1205902

119911119871

(64)


1205902

119911

1205902

119911119871

=120583119879

120583eq=

radic1 + 121205721205902

119911119871

minus 1

61205721205902

119911119871

(65)


120583eq = 120583119879[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

(66)


1205902

119911119871

=1205871198780

2120577119879

1205963

119899

=1205871198780

120583119879

1205962

119899

(67)



119911

1205902

119911

(1 + 31205781205902

119911

)

120583eq

120583119879

= 1205902

119911119871

(68)


1205902

119911

1205902

119911119871

= (radic1 + 12119871 minus 1

6119871)120583119879

120583eq(69)


1205962

eq = 1205962

119899

[1 +radic1 + 12119871 minus 1

2] (70)

where

119871 = 1205781205902

119911119871

120583119879

120583eq= 1205781205902

119911119871

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

minus1

(71)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

(72)





120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)


119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)




0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


005 s2mminus2







002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1205962

eq = 1205962

119899

(1 + 31205781205902

119911

) (38)


1205902

119911

= int

infin

minusinfin

119878119911

(120596) 119889120596 = 1198780

int

infin

minusinfin

|119867 (119894120596)|2

119889120596 (39)


119867(119894120596) =minus1

minus1205962

+ 1198942120577eq119879

120596eq120596 + 1205962

eq(40)


1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

2120577119879

120596119899

1205962

eq (41)


119911

31205781205904

119911

+ 1205902

119911

= 1205902

119911Lin=

1205871198780

2120577119879

1205963

119899

(42)

where 120590119911Lin



1205902

119911

=

radic1 + 121205781205902

119911Linminus 1

6120578

(43)


1205962

eq = 1205962

119899

[[

[

1 +

radic1 + 121205781205902

119911Linminus 1

2

]]

]

(44)


1

119866(119877119871

+ 119877119862

119877119871

)[ddt119881119871

(119905) + 2120577eq119879

120596eq119881119871 (119905) + 1205962

eq int119881119871 (119905) dt]

= minus 119910 (119905)

(45)


119867119881

(119894120596) = (119877119871

119877119871

+ 119877119862

)119866120596

(1205962

eq minus 1205962

) 119894 minus 2120577eq119879

120596eq120596 (46)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

times 1198662

1198780

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

= (119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times ((120596

1205962

eq)

2


120596eq)

2

]

2

+ [(120596119899

120596eq)(

1

119876119898

+1198662

(119898120596119899

(119877119871

+ 119877119862

)))

times (120596

120596eq)]

2

)

minus1

)

(47)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(48)


119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq119879

120596eq(49)



119903



119898





0

05

1

15

2

25

3




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)





+ 119887119879

120572119863()


119898 + [119887119879

+ 119887119879

120572119863 ()] + [119896119911 + 119896120578119873 (119911)] = minus119898 119910 (50)

0

1

2

3

4


times10minus4



SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)



+ 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

) = minus 119910 (51)


+ 119887119879

1205723



119879




120596119899

( + 1205723


(119911 + 1205781199113

)



+ 120583eq + 1205962

eq119911 = minus 119910 (52)


119890 = 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113


eq119911 (53)


120597

1205971205962

eq119864 [1198902

] = 0

120597

120597120583eq119864 [1198902

] = 0

(54)



119864 [2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [2

] minus 1205962

eq119864 [119911] = 0

119864 [1199112120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [119911] minus 1205962

eq119864 [1199112

] = 0

(55)



2

119911

and 119864[2] = 1205902

119911


120583eq =119864 [2120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(56)



1205962

eq =119864 [1199112120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(57)



120583eq = 119864[120597

120597[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

1205962

eq = 119864[120597

120597119911[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

(58)


120583eq = 119864 [2120577119879120596119899 (1 + 31205722

)] = 120583119879

(1 + 31205721205902

119911

) (59)


= 2120577119879

120596119899


1205962

eq = 119864 [1205962

119899

(1 + 31205781199112

)] = 1205962

119899

(1 + 31205781205902

119911

) (60)




1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

120583eq1205962

eq(61)


1205902

119911

=1205871198780

2120577eq119879

120596eq=1205871198780

120583eq(62)



1205902

119911119871

=1205871198780

2120577119879120596119899

=1205871198780

120583119879

(63)



119911

1205902

119911

(1 + 31205721205902

119911

) = 1205902

119911119871

(64)


1205902

119911

1205902

119911119871

=120583119879

120583eq=

radic1 + 121205721205902

119911119871

minus 1

61205721205902

119911119871

(65)


120583eq = 120583119879[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

(66)


1205902

119911119871

=1205871198780

2120577119879

1205963

119899

=1205871198780

120583119879

1205962

119899

(67)



119911

1205902

119911

(1 + 31205781205902

119911

)

120583eq

120583119879

= 1205902

119911119871

(68)


1205902

119911

1205902

119911119871

= (radic1 + 12119871 minus 1

6119871)120583119879

120583eq(69)


1205962

eq = 1205962

119899

[1 +radic1 + 12119871 minus 1

2] (70)

where

119871 = 1205781205902

119911119871

120583119879

120583eq= 1205781205902

119911119871

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

minus1

(71)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

(72)





120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)


119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)




0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


005 s2mminus2







002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












0

05

1

15

2

25

3




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)





+ 119887119879

120572119863()


119898 + [119887119879

+ 119887119879

120572119863 ()] + [119896119911 + 119896120578119873 (119911)] = minus119898 119910 (50)

0

1

2

3

4


times10minus4



SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)



+ 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

) = minus 119910 (51)


+ 119887119879

1205723



119879




120596119899

( + 1205723


(119911 + 1205781199113

)



+ 120583eq + 1205962

eq119911 = minus 119910 (52)


119890 = 2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113


eq119911 (53)


120597

1205971205962

eq119864 [1198902

] = 0

120597

120597120583eq119864 [1198902

] = 0

(54)



119864 [2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [2

] minus 1205962

eq119864 [119911] = 0

119864 [1199112120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [119911] minus 1205962

eq119864 [1199112

] = 0

(55)



2

119911

and 119864[2] = 1205902

119911


120583eq =119864 [2120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(56)



1205962

eq =119864 [1199112120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(57)



120583eq = 119864[120597

120597[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

1205962

eq = 119864[120597

120597119911[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

(58)


120583eq = 119864 [2120577119879120596119899 (1 + 31205722

)] = 120583119879

(1 + 31205721205902

119911

) (59)


= 2120577119879

120596119899


1205962

eq = 119864 [1205962

119899

(1 + 31205781199112

)] = 1205962

119899

(1 + 31205781205902

119911

) (60)




1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

120583eq1205962

eq(61)


1205902

119911

=1205871198780

2120577eq119879

120596eq=1205871198780

120583eq(62)



1205902

119911119871

=1205871198780

2120577119879120596119899

=1205871198780

120583119879

(63)



119911

1205902

119911

(1 + 31205721205902

119911

) = 1205902

119911119871

(64)


1205902

119911

1205902

119911119871

=120583119879

120583eq=

radic1 + 121205721205902

119911119871

minus 1

61205721205902

119911119871

(65)


120583eq = 120583119879[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

(66)


1205902

119911119871

=1205871198780

2120577119879

1205963

119899

=1205871198780

120583119879

1205962

119899

(67)



119911

1205902

119911

(1 + 31205781205902

119911

)

120583eq

120583119879

= 1205902

119911119871

(68)


1205902

119911

1205902

119911119871

= (radic1 + 12119871 minus 1

6119871)120583119879

120583eq(69)


1205962

eq = 1205962

119899

[1 +radic1 + 12119871 minus 1

2] (70)

where

119871 = 1205781205902

119911119871

120583119879

120583eq= 1205781205902

119911119871

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

minus1

(71)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

(72)





120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)


119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)




0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


005 s2mminus2







002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119864 [2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [2

] minus 1205962

eq119864 [119911] = 0

119864 [1199112120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

minus 120583eq119864 [119911] minus 1205962

eq119864 [1199112

] = 0

(55)



2

119911

and 119864[2] = 1205902

119911


120583eq =119864 [2120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(56)



1205962

eq =119864 [1199112120577

119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]

1205902

119911

(57)



120583eq = 119864[120597

120597[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

1205962

eq = 119864[120597

120597119911[2120577119879

120596119899

( + 1205723

) + 1205962

119899

(119911 + 1205781199113

)]]

(58)


120583eq = 119864 [2120577119879120596119899 (1 + 31205722

)] = 120583119879

(1 + 31205721205902

119911

) (59)


= 2120577119879

120596119899


1205962

eq = 119864 [1205962

119899

(1 + 31205781199112

)] = 1205962

119899

(1 + 31205781205902

119911

) (60)




1205902

119911

=1205871198780

2120577eq119879

1205963

eq=

1205871198780

120583eq1205962

eq(61)


1205902

119911

=1205871198780

2120577eq119879

120596eq=1205871198780

120583eq(62)



1205902

119911119871

=1205871198780

2120577119879120596119899

=1205871198780

120583119879

(63)



119911

1205902

119911

(1 + 31205721205902

119911

) = 1205902

119911119871

(64)


1205902

119911

1205902

119911119871

=120583119879

120583eq=

radic1 + 121205721205902

119911119871

minus 1

61205721205902

119911119871

(65)


120583eq = 120583119879[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

(66)


1205902

119911119871

=1205871198780

2120577119879

1205963

119899

=1205871198780

120583119879

1205962

119899

(67)



119911

1205902

119911

(1 + 31205781205902

119911

)

120583eq

120583119879

= 1205902

119911119871

(68)


1205902

119911

1205902

119911119871

= (radic1 + 12119871 minus 1

6119871)120583119879

120583eq(69)


1205962

eq = 1205962

119899

[1 +radic1 + 12119871 minus 1

2] (70)

where

119871 = 1205781205902

119911119871

120583119879

120583eq= 1205781205902

119911119871

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

minus1

(71)


(120596) = 1198780


119878119881119871(120596) = (

119877119871

119877119871

+ 119877119862

)

2

1198662

1198780

times

(1205961205962

eq)2

[1 minus (120596120596eq)2

]

2

+ [2120577eq119879

(120596120596eq)]2

(72)





120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)


119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)




0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


005 s2mminus2







002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













120577eq119879

=120596119899

120577119879

120596eq

[[

[

1 +

radic1 + 121205721205902

119911119871

minus 1

2

]]

]

=120596119899

120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 + 12120572120590

2

119911119871

minus 1)]

=120596119899

2120596eq[1

119876119898

+1198662

119898120596119899

(119877119871

+ 119877119862

)]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

(73)


119871 = 1205781205871198981198780

119876119898

(119877119871

+ 119877119862

)

1205962

119899

[119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

]

times [1 +1

2(radic1 +

121205871205721198981198780

119876119898

(119877119871

+ 119877119862

)

119898120596119899

(119877119871

+ 119877119862

) + 1198662

119876119898

minus 1)]

minus1

(74)


1198812

119871

= 1198780

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(75)

and the mean power

119875119871

=1198780

119877119871

(119877119871

119877119871

+ 119877119862

)

2

1198662

120587

2120577eq120596eq(76)




0

05

1

15

2

25

3



times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


005 s2mminus2







002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








References














































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










002040608

1121416




times10minus4

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


002040608

1121416




times10minus6

SD o

f loa

d vo

ltage

(Vrm

s2r

ads

)


5 s2mminus2


119860



3141

3143

3145

3147

0 002 004 006


Equi

vale

nt fr

eque

ncy

(rad

s)

120572 = 001 s2 mminus2

120572 = 005 s2 mminus2

120572 = 01 s2 mminus2

120572 = 05 s2 mminus2

120572 = 10 s2 mminus2

120572 = 50 s2 mminus2


3

25

2

1

15

05

00 50 100 150 200

Mea

n po

wer

(W)


times10minus3

G2 = 001T2 m2

G2 = 01T2 m2G2 = 05T2 m2

G2 = 10T2 m2



5 Conclusions








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











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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