research article parameter analysis on torque...

Research ArticleParameter Analysis on Torque Stabilization forthe Eddy Current Brake A Developed Model Simulationand Sensitive Analysis

Quan Zhou123 Xuexun Guo123 Gangfeng Tan123 Xiaomeng Shen4

Yifan Ye3 and Zhaohua Wang3

1Hubei Key Laboratory of Advanced Technology for Automotive Components Wuhan University of Technology WuhanHubei 430070 China2Hubei Collaborative Innovation Center for Automotive Components Technology Wuhan Hubei 430070 China3School of Automotive Engineering Wuhan University of Technology Wuhan Hubei 430070 China4School of Economics Wuhan University of Technology Wuhan Hubei 430070 China

Correspondence should be addressed to Gangfeng Tan 18995570176189cn

Received 10 March 2015 Revised 15 May 2015 Accepted 17 May 2015

Academic Editor Emiliano Mucchi

Copyright copy 2015 Quan Zhou 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

Eddy current brake (ECB) is an attractive contactless brake whereas it suffers from braking torque attenuation when the rotatingspeed increases To stabilize the ECBrsquos torque generation property this paper introduces the concept of anti-magneto-motive forceto develop the ECB model on the fundamental of magnetic circles In the developed model the eddy current demagnetizationand the influence of temperature which make the braking torque attenuation are clearly presented Using the developed model ofECB the external and internal characteristics of the ECB are simulated through programming byMATLAB To find the sensibilityof the influences on ECBrsquos torque generation stability the stability indexes are defined and followed by a sensibility analysis onthe internal parameters of an ECB Finally this paper indicates that (i) the stability of ECBrsquos torque generating property could beenhanced by obtaining the optimal combination of ldquodemagnetization speed point and the nominal maximum braking torquerdquo (ii)The most remarkable influencing factor on the shifting the demagnetization speed point of ECB was the thickness of the air-gap(iii) The radius of pole shoersquos cross section area and the distance from the pole shoe center to the rotation center are both the mostsignificant influences on the nominal maximum braking torque

1 Introduction

Eddy current brake (ECB) is an attractive auxiliary brakingdevice for vehicles which could be directly controlled bywireComparing with some other auxiliary brake ECB enjoysthe following advantages [1ndash3] (a) the ECB is easy to becontrolled and it nearly does not suffer the braking delayfor it is directly controlled by the current applied in the wire(b)TheECB enjoys better performances in low-speed domainthan either regenerative brake or hydraulic retarder (c) TheECB does not require the internal combustion engine (ICE)to provide the negative pressure as the pneumatic source(nevertheless the hydraulic retarder and the engine brakeneed the internal combustion engine to provide negative

pressure when they are working) Overall the ECB shouldbe a promising auxiliary brake for future vehicles as vehiclestend to be more electric drove especially for the vehicleswithout internal combustion engine like pure electric vehiclesand fuel cell battery electric vehicles

For the ECB the applied current generates magnetic fluxin the core and an eddy current is induced around poleshoe of the core when the magnetic flux goes through arotating conductive disk The braking force is generated bythe interaction between eddy current and magnetic fluxAccording to the theoretic works in [4 5] the braking torqueof an ECB was simply expressed as a function of the angularspeed of a disk and the applied current when ignoring thedemagnetization If the angular speed is a constant the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 436721 10 pageshttpdxdoiorg1011552015436721

2 Mathematical Problems in Engineering

ShaftCoil

Core

Rotator

Pole shoe

Magnetic circle

Ra

r

d

A

C

Eddy current

x

z

y

z

lg

Pole shoersquosCross section area

Figure 1 Schematic of a typical ECB

braking force is proportional to the applied current and viceversa Actually the braking torque would not be continuouslyincreasing while the angular speed or applied current risesAccording to experimental results on ECBrsquos torque-speedproperty [2 6ndash8] at the very beginning the braking torqueincreases rapidly while the rotating speed ascends Neverthe-less after the braking torque peaked at its maximum valuethe braking torque tends to drop significantly if the rotatingspeed goes on increasing The reason of this phenomenon isthat the demagnetization effect cut down the ECBrsquos torquegeneration property while at the same time the ECBrsquos torquestabilization is destroyed Specifically for excitation eddycurrent brake to prevent the demagnetization effect notonly means to improve the braking stability but also to savethe energy To explore the electromagnetic property of theECB Smythe [1] firstly implies magnetic potential theoryto model the ECB and he described the demagnetizationeffects by deriving the Maxwellrsquos equations Wouterse [3]indicates structure index C based on the numerical analysisand experiment validation to simplified Smythersquos modelSimeu and Georgersquos firstly [4] introduced an ECB based onmagnetic circle theory and hismodel is widely used in controlalgorithm research on ECB [5 9] However Simeursquos modelcan only express how ECB retards the rotator in low-speedregion but it cannot fully describe the ECBrsquos property in high-speed region for it ignored the eddy current demagnetizationRecently with the development of computational sciencemany have tried to model ECB with FEM [10ndash12] althoughthe FEA results could be a reference for engineering design itis still reasonable to develop current ECBrsquos theoretical modelfor detecting the influences on ECBrsquos braking torque stability[6ndash8] Hence this paper presents a modelling method basedon magnetic circle theory which is easy to find the relationsof the internal parameters to the braking torque stabilizationUsing this developed ECB model the sensitivity of theinfluences of the braking torque stabilization could be easilyfound through simulation and related sensitivity analysis

This paper is organized as follows Section 2 presents thebasic construction of a typical ECB while the parameters

for modeling are present Section 3 introduces the modelingassumptions and the mathematical modeling works in thissection by introducing the concept of antimagnetic force adeveloped ECBmodel is derivate in detail Section 4 includesthe simulation results on both external parameters and inter-nal parameters of the ECB based on the developed modelIn Section 5 the braking torque stability index including thedemagnetization speed point (demagnetization speed point)and the nominal maximum braking torque is introducedfollowed by a sensibility analysis on internal parameters tobraking torque stability index

2 The Principle of ECB

The configuration of a typical ECB is presented in Figure 1A typical ECB consisted of rotator shaft coil and ironcore For easy fabrication this paper presents an ECB thatis designed with two pairs of coil-core systems which arecircumferentially equispaced around the rotator The rotatoris a copper made disc with the radius of 119877 and the thicknessof 119889 In each coil-core system two U-shaped iron cores aresymmetry arranged aside the rotatorThe zone between thesetwoU-shaped iron cores calls air-gap the thickness of the air-gap is 119897119892 At the heat of each iron core the coils are windinginto1198738 turns and each coil is series-wound connected Thepole shoe is at the top of the U-shaped iron-core and in thispaper the pole shoersquos cross section area is roundness withradius of 119903 and the area of 119860

Figure 2 displays how the eddy currents are generatedand how the ECB retards the vehicle The applied currentgeneratesmagnetic flux in the core (see the green dotted circlein Figure 1) and an eddy current (see the green solid circle inFigures 1 and 2) is induced around pole shoe (the pole areaof the core see Figure 1) of the core when the magnetic fluxgoes through a rotating conductive disk The braking force(its direction is negative to the rotating speed see Figure 2)is generated by the interaction between eddy current andmagnetic flux The braking torque generated by ECB is then

Mathematical Problems in Engineering 3

Eddy current

Braking torque

y

z


120596

Figure 2 Working principle of a typical ECB

transferred to the wheels through the driving shaft and finalreducing gear

3 Mathematical Modeling of the ECB

31 Simplifying Equivalent Magnetic Circle and AssumptionsAccording to Ampere-Maxwell equation the magnetic fieldintensity generated by the applied current could be calculatedas

∮119897

119867119889119897 = 119868net (1)

In one pair of the coil-corn systems the path of integration isa closed circle and hence in one pair of the coil-corn system∮119897119867119889119897 = 119867sdot119897 where 119897 is the equivalent length of themagnetic

circle and 119867 is the magnetic field intensity 119868net is the overallcurrent around the magnetic circle and it is calculated as119868net = 119873sdot119894 where119873 is the turns of the coil and 119894 is the appliedcurrent in the coil

When given the definition of magnetic flux density 119861 =

120583119867 the overall magnetic flux in one pair of the coil-cornsystem Φ is calculated as

Φ = 119861119860 =120583119873119894119860

119897 (2)

When defining themagnetic force asF = 119868net119860 as the cross-section area of the pole shoe and the reluctance asR = 119897120583119860the basic magnetic theory is established

Φ =F

R (3)

In (3) F is combined magnetic force which is the sumof magnetic force and antimagnetic force R is the overallreluctance which could be calculated as the overall resistancein electric circles

According to magnetic circle theory the equivalent mag-netic circle of ECB is established in Figure 3

The modeling work on ECB is based on its equivalentmagnetic circle for further derivation this paper presents thefollowing assumptions

(1) The air-gap is acceptable thick that the cross-sectionarea ofmagnetic flux in the rotator could be estimatedas the cross section area of the pole shoe namely119860 =

1205871199032

(2) In the rotator the reluctance in shaft direction isRlowast119889119886119909

= 11988912058301205871199032 while the reluctance in radius

direction isRlowast119889119903119886

= 1198861205830119903119889 Here assume that 119886 ≫ 119889so that Rlowast

119889119903119886

≫ Rlowast119889119886119909

and all the magnetic flux couldcross through rotator disc from the shaft direction

(3) The influence of the radius of the rotator disc 119877 isignored as this factor only affects the value of thebraking torque but not the stability of the brakingtorque and the influence of this factor on the brakingtorque is discussed in some presented works beforesee [1 3 13]

(4) The braking time is acceptable short that the temper-ature variation could be ignored

Hence the simplified equivalent magnetic circle of ECB (seeFigure 4) could be rebuild as a series-wound magnetic circlewith magnetic forceF119894 air-gap reluctanceR119886 antimagneticforce generated by eddy current in the rotatorFeddy and ironcore reluctanceR119888

32 Derivation of the ECBModel According to the simplifiedequivalent magnetic circle the magnetic flux in one coil-coresystem is calculated as

Φ119888 =

F119894 minusFeddy

R119888 +R119886 +R119889 (4)

whereF119894 is the magnetic force generated by applied currentsin the coil and Feddy is the defined antimagnetic forceprovided by eddy current in the rotator R119888 R119886 and R119889are the reluctance value of coil air-gap and rotator discrespectively

The reluctance of the air-gap could be calculated as

R119886 =119897119892 minus 119889

1205830119860 (5)

where 1205830 is the permeability of vacuum with the value 4120587 times

10minus7 119897119892 is the distance between two pole shoes and 119889 is the

thickness of rotator disc and119860 is the cross section area of thepole shoe

In this paper the rotator is made up of copper hence itspermeability is similar to the permeability of vacuum andhence we rewrote the reluctance outside the iron corn as

R119892 = R119886 +R119889 =119897119892

1205830119860 (6)

The reluctance of the core could be calculated as

R119888 =119897119888

1205831198881205830119860 (7)


Antimagnetic

Magnetic forcelowast


forcelowast

Antimagnetic forcelowast



+minus + minus + minus

+minus +minus

+minus

+ minus+ minus



Φc

ℛlowastd119886119909




ℛlowastg

ℛlowastg ℛlowast

g

ℛlowastc

ℛlowastg

ℛlowastc

Figure 3 Diagram of the equivalent magnetic circle of ECB

Φc

Magnetic force

Antimagnetic force

ℛg

ℛd

ℛc

Figure 4 Diagram of the simplified equivalent magnetic circle ofECB

In (7) 119897119888 is the length of the equivalent magnetic circle in ironcore120583119888 is relative permeability of ferromagneticmaterialThevalue of 120583119888 is around 6000 so that the value ofR119888 is so muchless thanR119892 and hence it could be neglected in (2)

Then (4) could be rewritten as

Φ119888 =

F119894 minusFeddy

R119892 (8)

where the magnetic force F119894 could be calculated by multi-plying the turns of coil119873 and the applied current in the coil119894

F119894 = 119873119894 (9)

In addition the defined antimagnetic force Feddy generatedby eddy current in the rotator disc could be calculated as

Feddy = ∮119904

119869119890 119889119904 = 119869119890 sdot 119889 sdot 119903 (10)

Here 119869119890 is the current density of eddy current in the rotatordisc

119869119890 = 120590 sdot 119886 sdot (120596 times119861119888) (11)

In (8) 119886 is the distance from the center of pole shoe area tothe rotating center and 119861119888 = Φ119888119860 = Φ119888120587119903

2 is the magneticflux density in one pair of the coil-core system

Hence (10) could be rewritten as

Feddy = 120590 sdot 119886 sdot (120596timesΦ119888

1205871199032) sdot 119889 sdot 119903

= 120590 sdot 119886 sdot 119889 sdot (120596timesΦ119888

120587119903)

(12)

According to (4) the antimagnetic force could be also calcu-lated by

Feddy = 119873119894 minus Φ119888 sdotR119892 (13)

From (12) and (13) the magnetic flux in one coil-core systemis obtained as

Φ119888 =119873119894

(R119892 + (120590119886119889120596120587119903))

(14)

Hence the magnetic flux density in this coil-core system is

119861119888 =Φ119888

1205871199032=

119873119894

((1198971198921205830) + 120590119886119889119903120596)

(15)


119869119890 = 120590 sdot 119886 sdot 120596 times(119873119894

(1198971198921205830) + 120590119886119889119903120596

) (16)

The total power dissipation may be calculated simply byintegrating 120588sdot119869119890

2 over the cylindrical volume 1205871199032 sdot119889 [3 14 15]where 119889 and 119903 denote respectively the disc thickness and theradius of a circlewith the same area as the pole faceThereforethe total dissipation is

119875diss = 120588 sdot 1198691198902sdot 120587119903

2sdot 119889 (17)


From view of energy follow Theory the dissipated energyin eddy current is equal to the dissipated kinetic energy ofvehicle and hence the braking torque could be evaluated as

119879119887 =119875diss120596

= 1205901198862120587119903

2119889(

119873

(1198971198921205830) + 120590119886119889119903120596

)

2

1198942120596 (18)

In addition 120590 is the conductivity of the rotor material and itcould be calculated as

120590 =1

1205880 (1 + 120572119905) (19)

Here 119905 is the temperature of the rotating disc 1205880 and 120572

are electrical resistivity at 0∘C and temperature coefficientof resistance respectively For a copper made rotator disc1205880 = 175 times 10

minus8Ωm and 120572 = 41 times 10

minus31∘C Hence (18)

could be developed into

119879119887 = 12058711990321198862119889(

1198731205830

11989711989212058802(1 + 120572119905)

2+ 1198861198891199031205961205880 (1 + 120572119905)

)

2

sdot 1198942120596

(20)

4 Simulation on ECB

41 Simulation on the External Characteristics of the ECBSimulations are performed on the developed mathematicalmodel of braking torque in ECB by means of MATLAB

For a typical commercial vehicle which is equipped withan ECB in its driving shaft the ECB rotatorrsquos rotating speed ishighly related to the vehicles driving speedTheir relationshipis 119899ECB = (Vvehilce119903wheel)1198940 where Vvehicle is the vehiclersquos drivingspeed 119903wheel is the radius of the rear-wheel and 1198940 is thedriving ratio of the final reducing gear According to Chinarsquostransportation laws in most highways the maximum speedof commercial vehicle is limited to 80 kmh and hence inthe simulation the rotatorrsquos rotating speed is set from 0 to2500 rpm Simulations on the braking torque of ECB werecarried out at the speed from 0 to 2500 rpm the appliedcurrent was set from 0 to 20A and the temperature of rotatordisc was set from 0 to 800∘C with the parameters employedin simulations listed in Table 1

Figure 5 shows the ECBrsquos torque versus rotating speedproperty when the current is kept constant in 10A and thetemperature of the rotator disc is 200∘C the simulation isbased on the assumption that the influence of temperaturetransient variation on the braking torque is ignored Thesimulation result indicates that the working condition ofan ECB could be divided into two domains namely low-speed domain and high-speed domain The properties inboth domains are acceptable corresponding to the benchtest results present in [1 13] In low-speed domain thebraking torque is proportional to the rotatorrsquos rotating speedNevertheless the proportional relationship is broken as thespeed increases There is a peak value on the curve and theabscissa of this peak value is a demagnetization speed pointThe ECB came to its high-speed domain when the rotating

Table 1 Model parameters of ECB

Parameter Value Unit119886 85 mm119889 3 mm119903 20 mm119873 720 turns119897119892

16 mm1205830 4120587 times 10

7 Hm1205880 175 times 10

minus8Ωm

120572 41 times 10minus3 1∘C

Rotating speed (rpm)

Brak

ing

torq

ue (N

m)

Lowspeed

domain

Highspeed

domain

Peak torque

Figure 5 Simulation result of torque-speed property of ECB

speed surpasses the demagnetization speed point and inhigh-speed domain the braking torque and the rotatingspeed have an inverse proportional relation

From the simulation result presented in Figure 6 both theapplied current and the rotating speed affected the brakingperformance of ECB In low-speed domain the brakingtorque tended to be increasing while applied current orrotating speed was arising Nevertheless when the rotatingspeed surpasses a nomination value the braking torqueattenuates dramatically and the reducing rate of the torqueattenuates is proportional to the applied current whichmeans if the braking torque starts to attenuate the brakingtorque could not be stalely controlled by simply changing theapplied current

The temperature also affects the torque-speed charac-teristics of ECB Figure 7 shows the braking torque char-acteristics versus both temperature and applied currentthe braking torque tended a significantly attenuation whilethe temperature of rotator disc is continuously increasingDeserve to be mentioned the developed model in this papermaybe circumscribed to fully detecting the influence oftemperature on the braking torque especially the barkingtorque characteristics versus both temperature and rotat-ing speed The further developing model should considerthe thermal dynamics and this paper just simply observes


0

10

20

0

1000

2000

0

20

40

60

Applied current (A)Rotating speed (rpm)

Brak

ing

torq

ue (N

m)

0 5 10 15 20 25 30 35 40

Figure 6 Simulation result of braking torque characteristics (rotat-ing speed and applied current)

0200

400600

800

1516

1718

192010

15

20

25

Applied current (A)

Brak

ing

torq

ue (N

m)

12 14 16 18 20 22

Temperature (∘C)

Figure 7 Simulation result of braking torque characteristics(applied current and temperature)

the braking torque characteristic versus temperature in theview of qualitative analysis

42 Simulation on the Internal Characteristics of the ECBAccording to the derived mathematical models of brakingtorque in ECB the influencing parameters on the ECBrsquosbraking torque stability are the radius of the pole shoersquoscross-section area distance from the pole shoersquos area centerto the rotating center thickness of the rotator disc thethickness of the air gap and so forth With variations of

0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(1500 54362)

(750 108723)

(600 135904)

(500 163085)

(1000 81543)

r = 10mmr = 15mmr = 20mm

r = 25mmr = 30mm

Figure 8 Influence of the equivalent radius of pole shoe area

parameters described above the influences on ECB torque-speed property were studied as the applied current is 10 Aand the rotator disc temperature is 200∘C

As described in Figure 8 the increasing ratio of brakingtorque in low-speed domain increased with the rise ofthe radius of the pole shoersquos cross section area while theattenuation ratio in high-speed domain is also increased withthe rise of the radius of the pole shoersquos cross section areaWhen the value of the radius of the pole shoersquos cross sectionarea increases the value of the peak torque is increasingwhereas the demagnetization speed point is declining

As shown in Figure 9 the ECBrsquos torque-speed enjoys asimilar tendency while changing the distant from the poleshoersquos center the rotating center

In Figure 10 the increasing ratio of braking torque in low-speed domain drops with the rise of the thickness of the airgap while the attenuation ratio in high-speed domain is alsodamping with the rise of the thickness of the air gap Whenthe value of the thickness of the air gap increases the value ofthe peak torque is descending whereas the demagnetizationspeed point is aggrandizing

Figure 11 indicates that in the low-speed domain theincreasing ratio of braking torque arises with the rise of thethickness of the rotator disc while the attenuation ratio inhigh-speed domain is also increasing with the rise of thethickness of the rotator disc While the value of the thicknessof the rotator disc is increasing the value of the peak torquedoes not change significantly whereas the demagnetizationspeed point is aggrandizing

Conclusively the simulations indicate that the internalparameters can affect the ECBrsquos torque generating propertyFor a typical ECB before the rotating speed surpasses anominal value the braking torque provided by the ECBis proportional to the rotating speed In this domain thetorque could be controlled stably by changing value of applied


0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m)

(850 95932)(800 102328)

(750 108723)(710 115119)

(670 121514)

a = 95mma = 80mma = 85mm

a = 90mma = 95mm

Figure 9 Influence of the location of the pole shoe

0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(470 173958)

(710 11597)

(940 86979)(1180 69583)

(1410 57986)

lg = 10mmlg = 15mmlg = 20mm

lg = 25mmlg = 30mm

Figure 10 Influence of different air-gap thickness

current namely when the applied current is increasing boththe value and the increasing ratio of ECBrsquos braking torquecan increase On the contrary when the rotating speed goeson increase the braking torque cannot be simply controlledby changing the applied current as a proportional principlenamely when the applied current is increasing the brakingtorque increases whereas the increasing ratio drops whilethe rotation speed arises Hence to enhance the stability ofthe braking torque performance it is should find an optimalcombination of ECBrsquos demagnetization speed point and thenominal maximum braking torque which is defined as thetorque stabilization indexes by optimizing the ECBrsquoS internalparameters

0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m) (7575758 108723)

(5808081 108713)(4545455 108723)

(3787879 108723)(3282828 108721)

d = 3mmd = 4mmd = 5mm

d = 6mmd = 7mm

Figure 11 Influence of different rotator disc thickness

5 Sensibility Analysis on Influences ofTorque Stabilization

To investigate the torque stabilization indexes this paperfirstly introduces the concept of demagnetization speedpoint and it is judged by the demagnetization speed pointwhich could be found as the root of the partial differentialequation followed

120597119879119881

120597120596= 0 (21)

Hence the demagnetization speed point is described as afunction of distance from pole shoe center to the rotationcenter 119886 thickness of the rotator disc 119889 thickness of theair-gap 119897119892 and the equivalent radius of the pole shoersquos crosssection area 119903

120596de =119897119892

1205830120590119886119889119903 (22)

In the demagnetization speed point the nominal maximumbraking torque of the ECB could be obtained and it could becalculated as

119879max = 120587119903211988621198891205880 (1+120572119905)(

1198731205830

119897119892 + 119886119889119903 (1198971198921205830119886119889119903))

2

1198942

sdot

119897119892

1205830119886119889119903

(23)

Owing to the structure design the thickness of the rotatordisc is highly 119889 related to the thickness of the air-gap 119897119892 andthey must follow 119889 lt 119897119892 hence in the sensitive analysis thispaper observed the influence of the parameter 119897 = 119897119892 minus 119889

According to the simulations in Section 4 both thedemagnetization speed point and the nominal torque affect


Table 2 Initial value setting for sensitive analysis

Parameters Initial value Unit1198860 90 mm1198890 3 mm1198970 13 mm1199030 20 mm

the ECBrsquos braking performance hence this paper introducesa two-dimensional braking torque stability index that ismadeup with demagnetization speed point and nominal torqueWith the purpose to investigate the effects of the parametersof ECB on braking torque stabilization the related sensitivityanalysis is carried out The sensitivities of braking torquestabilization indexes to the parametric variations of ECBweredetected using one parametric variation method with theobserved parameters including charging the radius of thepole shoersquos cross section area distance from the pole shoersquosarea center to the rotating center thickness of the rotatordisc the thickness of the air gap and so forth At eachmeasurement the selected parameter was increased by 5of the initial value while other parameters kept invariableThen the variations of demagnetization speed point andthe nominal maximum braking torque with the selectedparameter were investigated The sensitivity can be definedas [16]

119878 =

10038161003816100381610038161003816100381610038161003816

Δ1199101199100Δ1199091199090

10038161003816100381610038161003816100381610038161003816

(24)

where 119878 is the sensitivity of index to the selected parametersΔ119910 is the variation of indexΔ119909 is the variation of the selectedparameter 1199090 is the initial value of the parameters and 1199100 isthe initial value corresponding to the situation when 119909 = 1199090The larger the sensitivity value themore significant the effectsof parameter on the evaluation of demagnetization speedpoint and maximum braking torque

Using the initial value settings in Table 2 Figures 12 and13 display the sensitivities of ECB performance to variousparameters and the calculated values of the sensitivity arelisted in Tables 3 and 4 respectively

From Figures 12 to 13 and Tables 3 to 4 the followingconclusions could be obtained (i) In terms of the effectson shift the demagnetization speed point the influencingparameters with the sensibility ranked in a descending orderwere the thickness of the air-gap the radius of pole shoersquoscross section area the distance from the pole shoe center tothe rotation center and thickness of the rotator disc (ii) Interms of the nominal maximum braking torque the radiusof pole shoersquos cross section area and the distance from thepole shoe center to the rotating center both were of mostsignificant the thickness of the air-gapwas ranked the secondplace while the influence of the thickness of the rotator disccould be ignored

6 Conclusions

In the presented work a design scheme of ECB was intro-duced and the mathematical model of ECB was developed

075

1

125

Para

met

er se

nsiti

vity

0

10

20

0 5 10 15 20Change from nominal value ()

Parameter aParameter d

Parameter lParameter r

minus5minus10minus15minus20



Para

met

er se

nsiti

vity

Figure 12 Sensitive of demagnetization speed point

0 5 10 15 200

025

05

075

1

125

15

Change from nominal value ()

Para

met

er se

nsiti

vity




Figure 13 Sensitive of maximum braking torque

based on the fundamental of the magnetic circle theorywhile introducing the concept of antimagnetic force Fur-thermore simulations on the braking torque characteristicswere performed based on the developedECBmodel inwhichthe influencing factors of torque generation properties wereinvestigated by the variations of the parameters in ECBrsquosstructure design By introducing the torque stability indexnamely the demagnetization speed point and the nominalmaximum braking torque the performance and influencingfactors of ECB were studied and the results are listed asfollows

(i) The stability of ECBrsquos torque generation propertycould be enhanced by shifting the demagnetization


Table 3 Sensitivity of demagnetization speed point

minus20 minus15 minus10 minus5 5 10 15 20119886 12500 11765 11111 10526 09524 09091 08696 08333119889 10156 09559 09028 08553 07738 07386 07065 06771119897 49874 66491 99724 199424 199376 99676 66442 49826119903 49813 66431 99667 199368 199429 99727 66493 49875

Table 4 Sensitivity of maximum braking torque

minus20 minus15 minus10 minus5 5 10 15 20119886 10000 10000 10000 10000 10000 10000 10000 10000119889 77e minus 16 0 15e minus 15 30e minus 15 30e minus 15 30e minus 15 10e minus 15 77e minus 16119897 09701 09253 08844 08496 07808 07514 07242 06989119903 10000 10000 10000 10000 10000 10000 10000 10000

speed point and the ECBrsquos demagnetization speedpoint could be shifted by optimizing the internalparameters when design

(ii) The most remarkable influencing factor on the shift-ing the demagnetization speed point of ECB was thethickness of the air-gap followed by the radius of poleshoersquos cross section area the distance from the poleshoe center to the rotation center and thickness of therotator disc

(iii) Thenominalmaximumbraking torque of anECBalsocould be optimized by changing the internal param-eters The most influencing factor on the nominalmaximumbraking torquewas the radius of pole shoersquoscross section area and the distance from the pole shoecenter to the rotation center followed by the thicknessof the air-gap and thickness of the rotator disc

Conflict of Interests

The authors declare no conflict of interests

Acknowledgments

The authors gratefully acknowledge the Hubei Key Labo-ratory of Advanced Technology for Automotive Compo-nents Hubei Collaborative Innovation Centre for Automo-tive Components Technology and the ldquoPlan-Dawn (CHENGUANG JI HUA)rdquo Funding for Youth Scientific and Techni-cal Researchers byWuhan Government (2013072304010830)In addition the authors acknowledge the members fromenergy-plus innovation group for eco-vehicle technology ofWUT

References

[1] W R Smythe ldquoOn eddy currents in a rotating diskrdquo Transac-tions of the American Institute of Electrical Engineers vol 61 no9 pp 681ndash684 1942

[2] J G Oetzel ldquoEddy current retarderrdquo SAE Technical Paper 54-0236 SAE 1954

[3] J H Wouterse ldquoCritical torque and speed of eddy currentbrake with widely separated soft iron polesrdquo IEE Proceedings BElectric Power Applications vol 138 no 4 pp 153ndash158 1991

[4] E Simeu and D Georges ldquoModeling and control of an eddycurrent brakerdquoControl Engineering Practice vol 4 no 1 pp 19ndash26 1996

[5] K Lee and K Park ldquoOptimal robust control of a contactlessbrake system using an eddy currentrdquo Mechatronics vol 9 no6 pp 615ndash631 1999

[6] D-S Li W-L Yin and K Zhang ldquoInfluence factor of brakingtorque in electromagnetic liquid-cooled retarderrdquo Journal ofBeijing University of Technology vol 6 pp 831ndash836 2014

[7] R He C-Y Liu and F-Y Yi ldquoComputationmethod for thermalfield of eddy current retarder in automobilerdquo Journal of JiangsuUniversity vol 26 no 2 pp 117ndash120 2005

[8] C Y Liu K J Jiang and Y Zhang ldquoDesign and use of aneddy current retarder in an automobilerdquo International Journalof Automotive Technology vol 12 no 4 pp 611ndash616 2011

[9] L Barnes J Hardin and C A Gross ldquoAn eddy current brakingsystemrdquo in Proceedings of the 25th Southeastern Symposium onSystemTheory (SSST rsquo93) pp 58ndash62 1993

[10] K V Tatis A G Kladas and J A Tegopoulos ldquoGeometry opti-mization of solid rotor eddy current brake by using sensitivityanalysis and 3D finite elementsrdquo Journal of Materials ProcessingTechnology vol 161 no 1-2 pp 363ndash367 2005

[11] R K Srivastava and S Kumar ldquoAn alternative approach forcalculation of braking force of an eddy-current brakerdquo IEEETransactions on Magnetics vol 45 no 1 pp 150ndash154 2009

[12] J-Y Choi H-J Shin Y-S Park and S-M Jang ldquoTorqueanalysis of axial flux PM type eddy current brake based onanalytical field computationsrdquo in Proceedings of the Interna-tional Conference on Electrical Machines and Systems (ICEMSrsquo11) August 2011

[13] R He X Liu and C Liu ldquoBrake performance analysis of ABSfor eddy current and electrohydraulic hybrid brake systemrdquoMathematical Problems in Engineering vol 2013 Article ID979384 11 pages 2013

[14] O Bottauscio M Chiampi and A Manzin ldquoModeling analysisof the electromagnetic braking action on rotating solid cylin-dersrdquo Applied Mathematical Modelling vol 32 no 1 pp 12ndash272008


[15] A Younis K Karakoc Z Dong E Park and A SulemanldquoApplication of SEUMRE global optimization algorithm inautomotive magnetorheological brake designrdquo Structural andMultidisciplinary Optimization vol 44 no 6 pp 761ndash772 2011

[16] H Zhang XGuo L Xu et al ldquoParameters analysis of hydraulic-electrical energy regenerative absorber on suspension perfor-mancerdquo Advances in Mechanical Engineering vol 2014 ArticleID 836502 12 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


ShaftCoil

Core

Rotator

Pole shoe

Magnetic circle

Ra

r

d

A

C

Eddy current

x

z

y

z

lg


Figure 1 Schematic of a typical ECB

braking force is proportional to the applied current and viceversa Actually the braking torque would not be continuouslyincreasing while the angular speed or applied current risesAccording to experimental results on ECBrsquos torque-speedproperty [2 6ndash8] at the very beginning the braking torqueincreases rapidly while the rotating speed ascends Neverthe-less after the braking torque peaked at its maximum valuethe braking torque tends to drop significantly if the rotatingspeed goes on increasing The reason of this phenomenon isthat the demagnetization effect cut down the ECBrsquos torquegeneration property while at the same time the ECBrsquos torquestabilization is destroyed Specifically for excitation eddycurrent brake to prevent the demagnetization effect notonly means to improve the braking stability but also to savethe energy To explore the electromagnetic property of theECB Smythe [1] firstly implies magnetic potential theoryto model the ECB and he described the demagnetizationeffects by deriving the Maxwellrsquos equations Wouterse [3]indicates structure index C based on the numerical analysisand experiment validation to simplified Smythersquos modelSimeu and Georgersquos firstly [4] introduced an ECB based onmagnetic circle theory and hismodel is widely used in controlalgorithm research on ECB [5 9] However Simeursquos modelcan only express how ECB retards the rotator in low-speedregion but it cannot fully describe the ECBrsquos property in high-speed region for it ignored the eddy current demagnetizationRecently with the development of computational sciencemany have tried to model ECB with FEM [10ndash12] althoughthe FEA results could be a reference for engineering design itis still reasonable to develop current ECBrsquos theoretical modelfor detecting the influences on ECBrsquos braking torque stability[6ndash8] Hence this paper presents a modelling method basedon magnetic circle theory which is easy to find the relationsof the internal parameters to the braking torque stabilizationUsing this developed ECB model the sensitivity of theinfluences of the braking torque stabilization could be easilyfound through simulation and related sensitivity analysis

This paper is organized as follows Section 2 presents thebasic construction of a typical ECB while the parameters

for modeling are present Section 3 introduces the modelingassumptions and the mathematical modeling works in thissection by introducing the concept of antimagnetic force adeveloped ECBmodel is derivate in detail Section 4 includesthe simulation results on both external parameters and inter-nal parameters of the ECB based on the developed modelIn Section 5 the braking torque stability index including thedemagnetization speed point (demagnetization speed point)and the nominal maximum braking torque is introducedfollowed by a sensibility analysis on internal parameters tobraking torque stability index

2 The Principle of ECB

The configuration of a typical ECB is presented in Figure 1A typical ECB consisted of rotator shaft coil and ironcore For easy fabrication this paper presents an ECB thatis designed with two pairs of coil-core systems which arecircumferentially equispaced around the rotator The rotatoris a copper made disc with the radius of 119877 and the thicknessof 119889 In each coil-core system two U-shaped iron cores aresymmetry arranged aside the rotatorThe zone between thesetwoU-shaped iron cores calls air-gap the thickness of the air-gap is 119897119892 At the heat of each iron core the coils are windinginto1198738 turns and each coil is series-wound connected Thepole shoe is at the top of the U-shaped iron-core and in thispaper the pole shoersquos cross section area is roundness withradius of 119903 and the area of 119860

Figure 2 displays how the eddy currents are generatedand how the ECB retards the vehicle The applied currentgeneratesmagnetic flux in the core (see the green dotted circlein Figure 1) and an eddy current (see the green solid circle inFigures 1 and 2) is induced around pole shoe (the pole areaof the core see Figure 1) of the core when the magnetic fluxgoes through a rotating conductive disk The braking force(its direction is negative to the rotating speed see Figure 2)is generated by the interaction between eddy current andmagnetic flux The braking torque generated by ECB is then


Eddy current

Braking torque

y

z


120596





∮119897

119867119889119897 = 119868net (1)





Φ = 119861119860 =120583119873119894119860

119897 (2)


Φ =F

R (3)





1205871199032





119889119903119886

≫ Rlowast119889119886119909






Φ119888 =

F119894 minusFeddy

R119888 +R119886 +R119889 (4)



R119886 =119897119892 minus 119889

1205830119860 (5)





R119892 = R119886 +R119889 =119897119892

1205830119860 (6)


R119888 =119897119888

1205831198881205830119860 (7)


Antimagnetic



forcelowast





+minus +minus

+minus

+ minus+ minus



Φc





ℛlowastg


g

ℛlowastc

ℛlowastg

ℛlowastc


Φc

Magnetic force

Antimagnetic force

ℛg

ℛd

ℛc




Φ119888 =

F119894 minusFeddy

R119892 (8)


F119894 = 119873119894 (9)


Feddy = ∮119904

119869119890 119889119904 = 119869119890 sdot 119889 sdot 119903 (10)


119869119890 = 120590 sdot 119886 sdot (120596 times119861119888) (11)





1205871199032) sdot 119889 sdot 119903


120587119903)

(12)




Φ119888 =119873119894

(R119892 + (120590119886119889120596120587119903))

(14)


119861119888 =Φ119888

1205871199032=

119873119894

((1198971198921205830) + 120590119886119889119903120596)

(15)


119869119890 = 120590 sdot 119886 sdot 120596 times(119873119894

(1198971198921205830) + 120590119886119889119903120596

) (16)



119875diss = 120588 sdot 1198691198902sdot 120587119903

2sdot 119889 (17)



119879119887 =119875diss120596

= 1205901198862120587119903

2119889(

119873

(1198971198921205830) + 120590119886119889119903120596

)

2

1198942120596 (18)


120590 =1

1205880 (1 + 120572119905) (19)






119879119887 = 12058711990321198862119889(

1198731205830

11989711989212058802(1 + 120572119905)

2+ 1198861198891199031205961205880 (1 + 120572119905)

)

2

sdot 1198942120596

(20)

4 Simulation on ECB






16 mm1205830 4120587 times 10

7 Hm1205880 175 times 10

minus8Ωm

120572 41 times 10minus3 1∘C


Brak

ing

torq

ue (N

m)

Lowspeed

domain

Highspeed

domain

Peak torque






0

10

20

0

1000

2000

0

20

40

60


Brak

ing

torq

ue (N

m)

0 5 10 15 20 25 30 35 40


0200

400600

800

1516

1718

192010

15

20

25

Applied current (A)

Brak

ing

torq

ue (N

m)

12 14 16 18 20 22

Temperature (∘C)




0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(1500 54362)

(750 108723)

(600 135904)

(500 163085)

(1000 81543)

r = 10mmr = 15mmr = 20mm

r = 25mmr = 30mm









0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m)

(850 95932)(800 102328)

(750 108723)(710 115119)

(670 121514)

a = 95mma = 80mma = 85mm

a = 90mma = 95mm


0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(470 173958)

(710 11597)

(940 86979)(1180 69583)

(1410 57986)


lg = 25mmlg = 30mm



0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m) (7575758 108723)

(5808081 108713)(4545455 108723)

(3787879 108723)(3282828 108721)


d = 6mmd = 7mm




120597119879119881

120597120596= 0 (21)


120596de =119897119892

1205830120590119886119889119903 (22)


119879max = 120587119903211988621198891205880 (1+120572119905)(

1198731205830

119897119892 + 119886119889119903 (1198971198921205830119886119889119903))

2

1198942

sdot

119897119892

1205830119886119889119903

(23)







119878 =

10038161003816100381610038161003816100381610038161003816

Δ1199101199100Δ1199091199090

10038161003816100381610038161003816100381610038161003816

(24)




6 Conclusions


075

1

125

Para

met

er se

nsiti

vity

0

10

20







Para

met

er se

nsiti

vity


0 5 10 15 200

025

05

075

1

125

15


Para

met

er se

nsiti

vity

















Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Eddy current

Braking torque

y

z


120596





∮119897

119867119889119897 = 119868net (1)





Φ = 119861119860 =120583119873119894119860

119897 (2)


Φ =F

R (3)





1205871199032





119889119903119886

≫ Rlowast119889119886119909






Φ119888 =

F119894 minusFeddy

R119888 +R119886 +R119889 (4)



R119886 =119897119892 minus 119889

1205830119860 (5)





R119892 = R119886 +R119889 =119897119892

1205830119860 (6)


R119888 =119897119888

1205831198881205830119860 (7)


Antimagnetic



forcelowast





+minus +minus

+minus

+ minus+ minus



Φc





ℛlowastg


g

ℛlowastc

ℛlowastg

ℛlowastc


Φc

Magnetic force

Antimagnetic force

ℛg

ℛd

ℛc




Φ119888 =

F119894 minusFeddy

R119892 (8)


F119894 = 119873119894 (9)


Feddy = ∮119904

119869119890 119889119904 = 119869119890 sdot 119889 sdot 119903 (10)


119869119890 = 120590 sdot 119886 sdot (120596 times119861119888) (11)





1205871199032) sdot 119889 sdot 119903


120587119903)

(12)




Φ119888 =119873119894

(R119892 + (120590119886119889120596120587119903))

(14)


119861119888 =Φ119888

1205871199032=

119873119894

((1198971198921205830) + 120590119886119889119903120596)

(15)


119869119890 = 120590 sdot 119886 sdot 120596 times(119873119894

(1198971198921205830) + 120590119886119889119903120596

) (16)



119875diss = 120588 sdot 1198691198902sdot 120587119903

2sdot 119889 (17)



119879119887 =119875diss120596

= 1205901198862120587119903

2119889(

119873

(1198971198921205830) + 120590119886119889119903120596

)

2

1198942120596 (18)


120590 =1

1205880 (1 + 120572119905) (19)






119879119887 = 12058711990321198862119889(

1198731205830

11989711989212058802(1 + 120572119905)

2+ 1198861198891199031205961205880 (1 + 120572119905)

)

2

sdot 1198942120596

(20)

4 Simulation on ECB






16 mm1205830 4120587 times 10

7 Hm1205880 175 times 10

minus8Ωm

120572 41 times 10minus3 1∘C


Brak

ing

torq

ue (N

m)

Lowspeed

domain

Highspeed

domain

Peak torque






0

10

20

0

1000

2000

0

20

40

60


Brak

ing

torq

ue (N

m)

0 5 10 15 20 25 30 35 40


0200

400600

800

1516

1718

192010

15

20

25

Applied current (A)

Brak

ing

torq

ue (N

m)

12 14 16 18 20 22

Temperature (∘C)




0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(1500 54362)

(750 108723)

(600 135904)

(500 163085)

(1000 81543)

r = 10mmr = 15mmr = 20mm

r = 25mmr = 30mm









0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m)

(850 95932)(800 102328)

(750 108723)(710 115119)

(670 121514)

a = 95mma = 80mma = 85mm

a = 90mma = 95mm


0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(470 173958)

(710 11597)

(940 86979)(1180 69583)

(1410 57986)


lg = 25mmlg = 30mm



0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m) (7575758 108723)

(5808081 108713)(4545455 108723)

(3787879 108723)(3282828 108721)


d = 6mmd = 7mm




120597119879119881

120597120596= 0 (21)


120596de =119897119892

1205830120590119886119889119903 (22)


119879max = 120587119903211988621198891205880 (1+120572119905)(

1198731205830

119897119892 + 119886119889119903 (1198971198921205830119886119889119903))

2

1198942

sdot

119897119892

1205830119886119889119903

(23)







119878 =

10038161003816100381610038161003816100381610038161003816

Δ1199101199100Δ1199091199090

10038161003816100381610038161003816100381610038161003816

(24)




6 Conclusions


075

1

125

Para

met

er se

nsiti

vity

0

10

20







Para

met

er se

nsiti

vity


0 5 10 15 200

025

05

075

1

125

15


Para

met

er se

nsiti

vity

















Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Antimagnetic



forcelowast





+minus +minus

+minus

+ minus+ minus



Φc





ℛlowastg


g

ℛlowastc

ℛlowastg

ℛlowastc


Φc

Magnetic force

Antimagnetic force

ℛg

ℛd

ℛc




Φ119888 =

F119894 minusFeddy

R119892 (8)


F119894 = 119873119894 (9)


Feddy = ∮119904

119869119890 119889119904 = 119869119890 sdot 119889 sdot 119903 (10)


119869119890 = 120590 sdot 119886 sdot (120596 times119861119888) (11)





1205871199032) sdot 119889 sdot 119903


120587119903)

(12)




Φ119888 =119873119894

(R119892 + (120590119886119889120596120587119903))

(14)


119861119888 =Φ119888

1205871199032=

119873119894

((1198971198921205830) + 120590119886119889119903120596)

(15)


119869119890 = 120590 sdot 119886 sdot 120596 times(119873119894

(1198971198921205830) + 120590119886119889119903120596

) (16)



119875diss = 120588 sdot 1198691198902sdot 120587119903

2sdot 119889 (17)



119879119887 =119875diss120596

= 1205901198862120587119903

2119889(

119873

(1198971198921205830) + 120590119886119889119903120596

)

2

1198942120596 (18)


120590 =1

1205880 (1 + 120572119905) (19)






119879119887 = 12058711990321198862119889(

1198731205830

11989711989212058802(1 + 120572119905)

2+ 1198861198891199031205961205880 (1 + 120572119905)

)

2

sdot 1198942120596

(20)

4 Simulation on ECB






16 mm1205830 4120587 times 10

7 Hm1205880 175 times 10

minus8Ωm

120572 41 times 10minus3 1∘C


Brak

ing

torq

ue (N

m)

Lowspeed

domain

Highspeed

domain

Peak torque






0

10

20

0

1000

2000

0

20

40

60


Brak

ing

torq

ue (N

m)

0 5 10 15 20 25 30 35 40


0200

400600

800

1516

1718

192010

15

20

25

Applied current (A)

Brak

ing

torq

ue (N

m)

12 14 16 18 20 22

Temperature (∘C)




0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(1500 54362)

(750 108723)

(600 135904)

(500 163085)

(1000 81543)

r = 10mmr = 15mmr = 20mm

r = 25mmr = 30mm









0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m)

(850 95932)(800 102328)

(750 108723)(710 115119)

(670 121514)

a = 95mma = 80mma = 85mm

a = 90mma = 95mm


0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(470 173958)

(710 11597)

(940 86979)(1180 69583)

(1410 57986)


lg = 25mmlg = 30mm



0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m) (7575758 108723)

(5808081 108713)(4545455 108723)

(3787879 108723)(3282828 108721)


d = 6mmd = 7mm




120597119879119881

120597120596= 0 (21)


120596de =119897119892

1205830120590119886119889119903 (22)


119879max = 120587119903211988621198891205880 (1+120572119905)(

1198731205830

119897119892 + 119886119889119903 (1198971198921205830119886119889119903))

2

1198942

sdot

119897119892

1205830119886119889119903

(23)







119878 =

10038161003816100381610038161003816100381610038161003816

Δ1199101199100Δ1199091199090

10038161003816100381610038161003816100381610038161003816

(24)




6 Conclusions


075

1

125

Para

met

er se

nsiti

vity

0

10

20







Para

met

er se

nsiti

vity


0 5 10 15 200

025

05

075

1

125

15


Para

met

er se

nsiti

vity

















Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879119887 =119875diss120596

= 1205901198862120587119903

2119889(

119873

(1198971198921205830) + 120590119886119889119903120596

)

2

1198942120596 (18)


120590 =1

1205880 (1 + 120572119905) (19)






119879119887 = 12058711990321198862119889(

1198731205830

11989711989212058802(1 + 120572119905)

2+ 1198861198891199031205961205880 (1 + 120572119905)

)

2

sdot 1198942120596

(20)

4 Simulation on ECB






16 mm1205830 4120587 times 10

7 Hm1205880 175 times 10

minus8Ωm

120572 41 times 10minus3 1∘C


Brak

ing

torq

ue (N

m)

Lowspeed

domain

Highspeed

domain

Peak torque






0

10

20

0

1000

2000

0

20

40

60


Brak

ing

torq

ue (N

m)

0 5 10 15 20 25 30 35 40


0200

400600

800

1516

1718

192010

15

20

25

Applied current (A)

Brak

ing

torq

ue (N

m)

12 14 16 18 20 22

Temperature (∘C)




0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(1500 54362)

(750 108723)

(600 135904)

(500 163085)

(1000 81543)

r = 10mmr = 15mmr = 20mm

r = 25mmr = 30mm









0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m)

(850 95932)(800 102328)

(750 108723)(710 115119)

(670 121514)

a = 95mma = 80mma = 85mm

a = 90mma = 95mm


0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(470 173958)

(710 11597)

(940 86979)(1180 69583)

(1410 57986)


lg = 25mmlg = 30mm



0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m) (7575758 108723)

(5808081 108713)(4545455 108723)

(3787879 108723)(3282828 108721)


d = 6mmd = 7mm




120597119879119881

120597120596= 0 (21)


120596de =119897119892

1205830120590119886119889119903 (22)


119879max = 120587119903211988621198891205880 (1+120572119905)(

1198731205830

119897119892 + 119886119889119903 (1198971198921205830119886119889119903))

2

1198942

sdot

119897119892

1205830119886119889119903

(23)







119878 =

10038161003816100381610038161003816100381610038161003816

Δ1199101199100Δ1199091199090

10038161003816100381610038161003816100381610038161003816

(24)




6 Conclusions


075

1

125

Para

met

er se

nsiti

vity

0

10

20







Para

met

er se

nsiti

vity


0 5 10 15 200

025

05

075

1

125

15


Para

met

er se

nsiti

vity

















Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

10

20

0

1000

2000

0

20

40

60


Brak

ing

torq

ue (N

m)

0 5 10 15 20 25 30 35 40


0200

400600

800

1516

1718

192010

15

20

25

Applied current (A)

Brak

ing

torq

ue (N

m)

12 14 16 18 20 22

Temperature (∘C)




0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(1500 54362)

(750 108723)

(600 135904)

(500 163085)

(1000 81543)

r = 10mmr = 15mmr = 20mm

r = 25mmr = 30mm









0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m)

(850 95932)(800 102328)

(750 108723)(710 115119)

(670 121514)

a = 95mma = 80mma = 85mm

a = 90mma = 95mm


0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(470 173958)

(710 11597)

(940 86979)(1180 69583)

(1410 57986)


lg = 25mmlg = 30mm



0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m) (7575758 108723)

(5808081 108713)(4545455 108723)

(3787879 108723)(3282828 108721)


d = 6mmd = 7mm




120597119879119881

120597120596= 0 (21)


120596de =119897119892

1205830120590119886119889119903 (22)


119879max = 120587119903211988621198891205880 (1+120572119905)(

1198731205830

119897119892 + 119886119889119903 (1198971198921205830119886119889119903))

2

1198942

sdot

119897119892

1205830119886119889119903

(23)







119878 =

10038161003816100381610038161003816100381610038161003816

Δ1199101199100Δ1199091199090

10038161003816100381610038161003816100381610038161003816

(24)




6 Conclusions


075

1

125

Para

met

er se

nsiti

vity

0

10

20







Para

met

er se

nsiti

vity


0 5 10 15 200

025

05

075

1

125

15


Para

met

er se

nsiti

vity

















Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m)

(850 95932)(800 102328)

(750 108723)(710 115119)

(670 121514)

a = 95mma = 80mma = 85mm

a = 90mma = 95mm


0 500 1000 1500 2000 25000

5

10

15

20


Brak

ing

torq

ue (N

m)

(470 173958)

(710 11597)

(940 86979)(1180 69583)

(1410 57986)


lg = 25mmlg = 30mm



0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16


Brak

ing

torq

ue (N

m) (7575758 108723)

(5808081 108713)(4545455 108723)

(3787879 108723)(3282828 108721)


d = 6mmd = 7mm




120597119879119881

120597120596= 0 (21)


120596de =119897119892

1205830120590119886119889119903 (22)


119879max = 120587119903211988621198891205880 (1+120572119905)(

1198731205830

119897119892 + 119886119889119903 (1198971198921205830119886119889119903))

2

1198942

sdot

119897119892

1205830119886119889119903

(23)







119878 =

10038161003816100381610038161003816100381610038161003816

Δ1199101199100Δ1199091199090

10038161003816100381610038161003816100381610038161003816

(24)




6 Conclusions


075

1

125

Para

met

er se

nsiti

vity

0

10

20







Para

met

er se

nsiti

vity


0 5 10 15 200

025

05

075

1

125

15


Para

met

er se

nsiti

vity

















Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119878 =

10038161003816100381610038161003816100381610038161003816

Δ1199101199100Δ1199091199090

10038161003816100381610038161003816100381610038161003816

(24)




6 Conclusions


075

1

125

Para

met

er se

nsiti

vity

0

10

20







Para

met

er se

nsiti

vity


0 5 10 15 200

025

05

075

1

125

15


Para

met

er se

nsiti

vity

















Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article parameter analysis on torque...

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