investigation on electromagnetic models of high-speed...

Research ArticleInvestigation on Electromagnetic Models of High-SpeedSolenoid Valve for Common Rail Injector

Jianhui Zhao1 Liyun Fan1 Peng Liu1 Leonid Grekhov2 XiuzhenMa1 and Enzhe Song1

1College of Power and Energy Engineering Harbin Engineering University Harbin 150001 China2College of Power Engineering Bauman State Technical University Moscow 115569 Russia

Correspondence should be addressed to Liyun Fan fanly_01163com

Received 26 December 2016 Revised 10 April 2017 Accepted 20 April 2017 Published 12 June 2017

Academic Editor Francesco Braghin

Copyright copy 2017 Jianhui Zhao 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

A novel formula easily applied with high precision is proposed in this paper to fit the 119861-119867 curve of soft magnetic materials and itis validated by comparison with predicted and experimental results It can accurately describe the nonlinear magnetization processand magnetic saturation characteristics of soft magnetic materials Based on the electromagnetic transient coupling principlean electromagnetic mathematical model of a high-speed solenoid valve (HSV) is developed in Fortran language that takes thesaturation phenomena of the electromagnetic force into consideration The accuracy of the model is validated by the comparisonof the simulated and experimental static electromagnetic forces Through experiment it is concluded that the increase of the drivecurrent is conducive to improving the electromagnetic energy conversion efficiency of the HSV at a low drive current but it haslittle effect at a high drive current Through simulation it is discovered that the electromagnetic energy conversion characteristicsof the HSV are affected by the drive current and the total reluctance consisting of the gap reluctance and the reluctance of theiron core and armature soft magnetic materials These two influence factors within the scope of the different drive currents havedifferent contribution rates to the electromagnetic energy conversion efficiency

1 Introduction

The fuel system has a vital impact on the overall performanceof a diesel engine and a high-pressure common rail systemenables the cyclic injection quantity injection timing andfuel injection law to be controlled exactly and flexiblyTherefore a diesel engine that is equipped with a high-pressure common rail system has the potential to achievethe optimized design goals of high-efficiency combustion andultralow emission [1ndash5] The common rail injector is one ofthe high-pressure common rail systemrsquos critical componentswith a direct influence on the cycle fuel injection quantityfluctuation gas-liquid two-phase flow characteristics of high-pressure fuel in the nozzle fuel atomization characteristicsin the cylinder and air-fuel mixture quality [6ndash9] The com-mon rail injector is a complex nonlinear and multidimen-sional system that couples electromagnetic mechanical andhydraulic phenomena To study the dynamic characteristicsof the common rail injector in detail and optimize it the

simulation analysis of its dynamic performance is an effectivemethod A simulation model of the HSV that has highaccuracy is thus indispensable

The dynamic response characteristic is an importantevaluation index of the HSV in the fuel injector This isbecause a fast response speed of the HSV is beneficialto achieve multiple injections and more precise control ofthe fuel injection timing of the high-pressure common railsystem The dynamic response of the HSV is determined byboth its electromagnetic force characteristic andmoving partquality but the former has a more significant impact Onlywhen the maximum static electromagnetic force of the HSVmeets its requirements can the response time requirements ofthe opening and closing phases of the HSV be met by usingthe drive circuit Hence research on the electromagneticforce is of great importance for the design of the HSV

In research on the static electromagnetic force of theHSV most scholars used the finite element method (FEM)Liu et al [10] studied the influence rules of the main

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 9078598 10 pageshttpsdoiorg10115520179078598

2 Mathematical Problems in Engineering

structure parameters and the interaction between them onthe electromagnetic force using a combination of responsesurface design and FEM Sun et al [11] studied the influencerules of the iron core length the cross-sectional areas of themain and side poles the coil turns and the air gap widthon the electromagnetic force of an E-shaped HSV for anelectronic unit pump by FEM It was discovered that the drivecurrent significantly impacts the electromagnetic energyconversion as increasing the drive current unnecessarilydoes not improve the electromagnetic force of the HSV butonly increases the power consumption and thus reducesthe electromagnetic energy conversion efficiency Cheng etal [12] studied using FEM the flux density distribution ofan HSV whose iron core is made of a nano-based softmagneticmaterial It was discovered that solenoid valves withdifferent softmagneticmaterials have different characteristicsof electromagnetic force Miller et al [13] Shin et al [14]and Bianchi et al [15] carried out optimization research onthe static electromagnetic force of an HSV to obtain theoptimal structure parameters by FEM However because ofthe long solution time and high requirements for computerperformance of FEM the efficiency of research on theoptimization and analysis of the structural parameters is lowespecially when carrying out an iterative structural analysisfor the detailed parameter design of HSV Therefore thedevelopment of a static electromagnetic mathematical modelfor an HSV that can be applied conveniently with highaccuracy has become an important research direction

Elmer and Gentle [16] put forward a relatively simplemathematical model of a proportional solenoid valve Themodel simplified the electromagnetic process calculation ofthe solenoid valve as the calculation of an RL equivalentcircuit Ma et al [17] Shamdani et al [18] and Bianchi et al[19] determined the influencing relation of drive current andair gap on electromagnetic force by a polynomial fit techniquebased on test data of the static electromagnetic force Wanget al [20] and Chung et al [21] considered the magnetic sat-uration characteristics of an HSV by adjusting the coefficientof the fitting formula to limit the maximum electromagneticforce of the valve Huber and Ulbrich [22] and Chung etal [23] developed a dynamic characteristic mathematicalmodel for the common rail injector Its electromagneticsubmodel was built based on a map chart consisting of theelectromagnetic force drive current and air gap HowevertheHSV electromagnetic characteristics are not only a simplefunction relationship between the electromagnetic force andthe drive current and air gap In the high-pressure commonrail system the nonlinear transient coupling characteristicsof the electromagnetic mechanical and hydraulic systemsdetermine the electromagnetic characteristic transformationof the HSV Therefore this method meets the demands ofengineering application but it cannot reveal the intrinsicelectromagnetic properties of the HSV both the static elec-tromagnetic characteristics and the dynamic electromagneticcharacteristics

Most of the literature on the HSV tends to ignorethe influence of the soft magnetic material reluctance indeveloping the electromagnetic model of the solenoid valveTopcu et al [24] Jin et al [25] Naseradinmousavi and

Nataraj [26] and Mehmood et al [27] considered that thereluctance of the iron core magnetic material is much lessthan that of the air gap therefore the nonlinear mag-netization process and magnetic saturation characteristicsof the iron core magnetic material can be ignored Sefkat[28] assumed that the magnetic field is not saturated inthe whole working process of the solenoid valve and themagnetic induction intensity always linearly increases withthe increasing magnetic field intensity that is to say thesaturation phenomenon of the electromagnetic force did nothappen for their model However Wang et al [29] and Sunet al [11] discovered that a saturation phenomenon of theelectromagnetic force for the solenoid valve existed in theirexperimental research and the magnetic saturation of theHSV soft magnetic materials was the main cause of this Inthe actual control process of the HSV to improve the openingspeed a high voltagewas loaded on theHSV to quickly obtaina large current In this way it is easy to achieve magneticsaturation for the solenoid valve When the solenoid valvereached magnetic saturation with the increase in the drivecurrent the electromagnetic force of the HSV will increasevery slowly therefore the economy of the HSV decreasedSimply dealing with the magnetic saturation characteristic ortaking no account of it will lead to obvious calculation errorsin determining the electromagnetic force characteristicsTherefore it is necessary to take the nonlinear magnetizationcharacteristics and magnetic saturation characteristics of thesolenoid valvemagneticmaterials into account upon buildinga mathematical model of an HSV

Coppo et al [30] established a complete mathematicalmodel for a common rail injector For its electromagneticsubmodels the magnetization process of the solenoid valvewas divided into a linear magnetization phase and a satu-ration magnetization phase which are described by simplepiecewise lines In the mathematical model of an HSVdeveloped by Liu et al [31] the reluctance of the softmagneticmaterials was included to consider the nonlinear magnetiza-tion characteristic but the most important 119861-119867magnetizingcurve was determined based on 119861-119867 experimental data by atrial and error method for calculating the reluctance of softmagnetic materials The mathematical models of a solenoidvalve developed by Jin et al [25] and Vu et al [32] consideredthe magnetic saturation characteristics edge effect of theair gap and flux leakage but a detailed description of the119861-119867 magnetizing curve that determines the reluctance ofthe soft magnetic materials and calculates their transientpermeability was not provided

It can be found from the above references that a certainresearch insufficiency on the electromagnetic mathematicalmodel of the HSV still exists Some electromagnetic mathe-matical models use an electromagnetic force fitting formulabased on experimental data some models do not considerthe reluctance of the soft magnetic materials and someonly took segmental magnetic properties of the magneticmaterials into account Therefore to provide more valuableinformation on the electromagnetic energy conversion ofthe HSV and more deeply understand the solenoid valversquoselectromagnetic conversion process more detailed researchon the electromagnetic mathematical model of the HSV

Mathematical Problems in Engineering 3

Coil Coil

Armature

Iron core

r1

r2

r3

r4

r5

r6

r8r7

la

lb

lc

SＣＨ SＩＯＮ

ld

h

Figure 1 The structure schematic of HSV

has been carried out in this paper The resulting modelconsiders the nonlinear magnetization characteristics andmagnetic saturation characteristics of the HSV Based on thismathematical model research on the electromagnetic forcecharacteristics of the common rail injector HSV and its keyinfluence factors can follow It provides a certain theoreticalbasis and design tools for the design of HSVs

2 The Electromagnetic Model of HSV

21 Magnetic Circuit Model of HSV Figure 1 shows the struc-ture schematic of a HSV It mainly consists of an iron corecoil and armature To obtain a strong electromagnetic forcea soft magnetic material with a high saturation inductiondensity and low remanence is often applied for the iron coreand armature

For theHSV the functional relationship between the totalmagnetic flux and the total reluctance of the magnetic circuitis described as

Φ = 119873 sdot 119868119877total (1)

whereΦ is the total magnetic flux119873 is the coil turns 119868 is thecurrent 119877total is the total reluctance of magnetic circuit

To accurately describe the nonlinear magnetization andmagnetic saturation of the HSV the mathematical modelshould take the reluctance of the iron core and armature softmagnetic materials into consideration Due to the differentequivalent cross-sectional areas of the main and side polesunder normal circumstances the gap reluctances between themain pole side pole and armature are described separatelyin the mathematical model and the fringe effects of the airgap and the leakage flux are ignored Based on the equivalentmagnetic circuit shown in Figure 2 the total reluctance of theHSV is expressed as

119877total = 119877gap1 + 119877gap2 + 119877arm + 119877iron (2)

where 119877gap1 is the gap reluctance between main pole andarmature 119877gap2 is the gap reluctance between side poleand armature 119877arm is the reluctance of armature 119877iron is

N middot I

RＡＪ1 RＡＪ2 RＬＧ

RＣＬＩＨ

Φ

Figure 2 Equivalent magnetic circuit of HSV

the reluctance of iron core Corresponding reluctances aredescribed as

119877gap1 = ℎ1205830 sdot 119878in (3)

119877gap2 = ℎ1205830 sdot 119878out (4)

119877arm = 11990382 sdot 120583 sdot 119878in +11990382 sdot 120583 sdot 119878out +

119897119889120583 sdot 11987810158401015840 (5)

119877iron = 119897119887120583 sdot 119878in +119897119887120583 sdot 119878out +

119897119886120583 sdot 1198781015840 (6)

where 119878in is the equivalent cross-sectional area of the HSVmain pole 119878out is the equivalent cross-sectional area of theHSV side pole ℎ is the air gap of the HSV 119897119887 is the equivalentmagnetic circuit length of the iron core in the axial direction119897119886 is the equivalent magnetic circuit length of the iron corein the radial direction 119897119889 is the equivalent magnetic circuitlength of the armature in the radial direction 120583 is thepermeability of the soft magnetic material depending on the119861-119867 fundamental magnetizing curve 11987811 is the equivalentmagnetic flux area of the iron core in the radial directionand 11987822 is the equivalentmagnetic flux area of the armature inthe radial direction Each equivalent magnetic circuit lengthis expressed as

119897119887 = 1199031 + 11990322

119897119886 = 1199035 + 1199036 minus 1199033 minus 11990344 119897119889 = 119897119887

11987811 = 120587 sdot (1199031 minus 1199032) sdot 1199035 + 1199036 + 1199033 + 11990344

11987822 = 120587 sdot 1199038 sdot 1199035 + 1199036 + 1199033 + 11990344

119878in = 1205874 sdot (1199032

4minus 11990323)

119878out = 1205874 sdot (1199032

6minus 11990325)

(7)


Finally the electromagnetic force is computed by

119865mag = 1212060121205830 sdot 119878in +

1212060121205830 sdot 119878out

(8)

22 119861-119867 Magnetization Curve Model The permeability ofthe soft magnetic material shows its influence by two aspectsin the electromagnetic mathematical model of the HSV Onthe one hand the reluctance of the softmagnetic material hasan effect on the electromagnetic conversion and on the otherhand the drive current has an effect on the reluctance Theelectromagnetic coupling of the HSV is exactly implementedby the permeability of the soft magnetic material in theelectromagnetic force mathematical model Therefore theaccuracy of the 119861-119867 curve fitting formula directly impactsthe accuracy of the results predicted by the electromagneticmathematical model of the HSV

Jiles andAtherton [33] proposed the Jiles-Athertonmodel(J-A model) to describe the phenomena of magnetizationand magnetic hysteresis The model is widely used in theelectromagnetic simulation field such as for motors How-ever when applying the model five key parameters needbe determined by fitting based on test data with a complexmathematical method Leite et al [34] used a genetic algo-rithm to determine the five parameters and verified themusing the least mean square error between the experimentaland simulated results Jaafar [35] studied the influence of thefive parameters on the magnetization process and found thateach parameter significantly impacts the prediction accuracyof the magnetization and hysteresis mathematical modelsHence accurately determining the five parameters introducesdifficulty to the engineering application of the J-A modelTherefore a mathematical model of the 119861-119867 fundamentalmagnetizing curve of a soft magnetic material that can beeasily applied with high precision is very necessary

Chan et al [36] assumed that the hysteresis loop of amagnetic material is symmetrical with the 119861-119867 fundamentalmagnetizing curve and obtained the following mathematicalmodel for the fundamental magnetizing curve

119861+ = 119861119904 sdot 119867 + 1198671198881003816100381610038161003816119867 + 1198671198881003816100381610038161003816 + 119867119888 sdot (119861119904119861119903 minus 1)

119861minus = 119861119904 sdot 119867 minus 1198671198881003816100381610038161003816119867 minus 1198671198881003816100381610038161003816 + 119867119888 sdot (119861119904119861119903 minus 1)

119861 = 119861+ + 119861minus2

(9)

The one-dimensional commercial software AMESim [37]used (10) to describe the 119861-119867 fundamental magnetizingcurve for the electromagnetic calculation where coefficient119860 is determined by (11)

119861 = 1205830 (119867 +119872119904 sdot tanh(119867119860)) (10)

119860 = 119872119904(119861119904 (1205830 sdot 119867119904)) minus 1 (11)

0

05

1

15

2

25

0 5 10 15 20 25 30 35 40 45 50 55 60Magnetic field intensity (H kAm)

Mag

netic

indu

ctio

n in

tens

ity (B

T)

Experimental dataSimulated by (10)

Figure 3 Comparison between simulated results by formula pro-posed by Chan and experimental results of 119861-119867 curve

Simulated by (11) with A = 5000Simulated by (11) and (12)

Experimental data

002040608

112141618

222

Mag

netic

indu

ctio

n in

tens

ity (B

T)

5 10 15 20 25 30 35 40 45 50 55 600Magnetic field intensity (H kAm)

Figure 4 Comparison between simulated results by formula pro-posed by AMESim and experimental results of 119861-119867 curve

where119872119904 is the saturation magnetization 119861119903 is the residualmagnetic flux density 119861119904 is the saturation induction density119867119904 is the magnetic field intensity corresponding the satura-tion induction density119867119888 is the coercivity 1205830 is the constantpermeability of vacuum

As described above previous studies used different fit-ting formulas for the 119861-119867 fundamental magnetizing curvetherefore it is necessary to study the applicability of theabove formula to obtain the most consistent fitting formuladescribing the magnetization process of a soft magneticmaterial As shown in Figure 3 the equation proposedby Chan provides a good prediction at the stages of theinitial magnetization and magnetic saturation but providespoor prediction in the transient process from the initialmagnetization to the magnetic saturation which is near theinflection point of the curve

As shown in Figure 4 the results predicted by (10) and (11)cannot adequately describe the real magnetization process ofmagneticmaterials Equation (10) alone can provide relativelygood prediction by adjusting its coefficient 119860 (119860 is 5000 in


Magnetic intensity (H Am)

Bs

Mag

netic

indu

ctio

n in

tens

ity (B

T)

Figure 5 Schematic diagram of 119861-119867magnetization curve

the prevailing circumstance) but it also provides predictionin the transient process from the initial magnetization tomagnetic saturation

Figure 5 shows the 119861-119867 fundamental magnetizing curveof magnetic materials including hard magnetic materialsand soft magnetic materials With the increase of 119867 119861increases quickly at first but its increase becomes slowand gradually approaches the saturation induction density119861119904 Therefore the selected fitting formula can accuratelydescribe the physical phenomena of the 119861-119867 fundamentalmagnetizing curve

In numerous functional forms the curve of the logarith-mic function log(119909) comes close to the curve in Figure 5Additionally when 119867 is infinite the use of a logarithmicfunction can ensure that 119861 still continues to increase slightlywith the further increase of119867 This can make the permeabil-ity 120583 (120583 = Δ119861Δ119867) nonzero in the whole range of 119867 andensure that (5) and (6) have physical meaning Intending toreduce the number of coefficients in the fitting formula of the119861-119867 fundamental magnetizing curve the natural logarithmln(119909) is chosen as the basic functional form However foreach type ofmagneticmaterial the fundamentalmagnetizingcurve starts from the coordinate origin (0 0) thus when 119867is zero 119861 is zero As the natural logarithm ln(119909) does notpass through this point the equation must be changed to theform of ln(119909 + 1) thus the fitting formula will pass throughthe coordinate origin (0 0) Considering the above factorsthe basic functional form of the fitting formula of the 119861-119867fundamental magnetizing curve is determined as follows

119861 = ln (119867 + 1) (12)

Every magnetic material has a different saturation induc-tion density 119861119904 and maximum permeability 120583max 119861119904 deter-mines themagnetic saturation characteristics of themagneticmaterial and 120583max determines its nonlinear magnetizationprocess To ensure the universality of the proposed fittingformula two coefficients 1199011 and 1199012 are introduced 1199011 isapplied to adjust the saturation induction density 119861119904 and 1199012is applied to adjust the transformation process of the mag-netizing curve from the initial magnetization to the criticalmagnetic saturation Equation (13) introduces coefficient 1199011on the basis of (12) As shown in Figure 6 the saturationinduction density of the fitting curve gradually increases with

Simulated with p1 = 03 Simulated with p1 = 02Experimental data Simulated with p1 = 01


004081216

224283236

Mag

netic

indu

ctio

n in

tens

ity (B

T)

Figure 6 Influence of 1199011 on 119861-119867 fitting curve

Simulated with p2 = 00003Simulated with p2 = 00009

Simulated with p2 = 00001Simulated with p2 = 00006Experimental data

004081216

224283236

444

Mag

netic

indu

ctio

n in

tens

ity (B

T)



the increase in 1199011 When 1199011 is 02 the saturation inductiondensities of the simulation and experiment match especiallyclosely However no matter what the value of 1199011 is there is arelatively large error between the simulation and experimentin the initial stage of magnetization Hence the equationcannot describe the nonlinear magnetization process ofmagnetic materials in the initial stage of magnetization

119861 = 1199011 sdot ln (119867 + 1) (13)

Equation (14) introduces coefficient 1199012 on the basis of(12) As shown in Figure 7 with the increase of 1199012 the fittingcurve and experimental data match more and more closelyin the initial stage of magnetization When 1199012 is 00009 (14)can particularly accurately describe the initial magnetizationprocess of the magnetic material However since 1199011 isnot introduced to limit the maximum magnetic inductionintensity the predicted saturation induction density willincrease with 1199012

119861 = ln (1199012 sdot 119867 + 1) (14)


Table 1 Measurement accuracy of main equipment

Equipment Force sensor Current probeType CZLYB-3 1146AProducer Chengdu Xingpu Transducer Co Ltd Agilent TechnologyMeasurement range 0sim500N 1sim100AMeasurement accuracy leplusmn005 leplusmn2

0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)



Figure 8 Comparison between simulated results by (15) andexperimental results of 119861-119867 curve

Simulated by (16)Experimental data

0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)

3510 15 5550305 40 45 600 2520Magnetic field intensity (H kAm)


It can be observed from the above analysis that the twocoefficients 1199011 and 1199012 should be introduced to accuratelydescribe the 119861-119867 fundamental magnetizing curve in (12)Equation (15) introduces1199011 and1199012 which can be determinedby least square method

119861 = 1199011 sdot ln (1199012 sdot 119867 + 1) (15)

As shown in Figure 8 there is a relatively remarkable errorin the magnetic saturation phase and the transition phasefrom the unsaturated regime to the saturated state betweenthe simulated results from (15) and the experimental resultsThe reason is that the growth rate of magnetic induction

intensity is relatively larger in the magnetic saturation phaseTo avoid this the square root of the logarithmic functionin (15) is used The ultimate fitting formula of the 119861-119867fundamental magnetizing curve is shown as follows

119861 = 1199011 sdot radicln (1199012 sdot 119867 + 1) (16)

As shown in Figure 9 the simulated results from (16)and the experimental results match closely in the wholemagnetization process and their 119877-square determinationcoefficient is 096 This proves that the proposed fittingformula can accurately describe not only the magneticsaturation phenomenon of magnetic materials but also thenonlinear magnetization process

3 Validation of ElectromagneticModel for HSV

31 Test Bench of Electromagnetic Force of HSV To experi-mentally validate the accuracy of the electromagnetic modelfor the HSV the test bench of the static electromagnetic forcedepicted in Figure 10 was used The iron core is placed at thebench free end and the armature and force sensor (ChengduXingpu Transducer Co Ltd CZLYB-3) are positioned atthe bench fixed end The height of the free end is adjustedsuch that the iron core axis and the armature axis are inthe same horizontal line The total air gap is changed byadjusting the distance between the free end and the fixedend The drive current of the solenoid is delivered by acurrent-controlled power amplifier and it is measured usinga current probe (Agilent Technology 1146A) The armatureis attracted towards the iron core after a constant current tothe coil is turned on and the force sensor generates a weakvoltage signalThis voltage signal represents themagnitude ofthe electromagnetic force in the axial direction after passingthrough the high-precision amplifier Experimental resultswere obtained by changing the air gap and drive current Themeasurement accuracy of the main equipment is shown inTable 1

32 Model Validation Table 2 shows the detailed structuralparameters of the HSV studied in this paper The iron coreand armature aremade using the same softmagneticmaterialwhose 119861-119867 fundamental magnetizing curve is shown in Fig-ure 11 The air gap between the armature and electromagnetis adjusted to 01mm and 012mm in the test bed of the staticcharacteristics of the HSV The coil of the HSV is energizedto different drive currents for the above working air gapsand the static electromagnetic forces of the HSV at different


Bench fixed end

Amplifier Oscilloscope

Current probe

Driving system

Bench free end

Armature

Iron core

Force sensor

(a) The schematic block diagram of the test bench

Bench fixed end Bench free end

Currentprobe

Electromagneticvalve

Amplifier

Force sensor

Driv

ing

syste

m

(b) Test apparatus

Figure 10 Test bench of electromagnetic force for HSV

Table 2 Structural parameters of HSV

Parameters Reference valueIron core

Height 1199031 (mm) 137Diameter 1199036 (mm) 204Diameter of hole 1199033 (mm) 7119861-119867 curve From Figure 11

CoilNumber of turns119873 52Height 1199032 (mm) 76Inside diameter 1199034 (mm) 12Outside diameter 1199035 (mm) 177

ArmatureThickness 1199038 (mm) 18Diameter 1199037 (mm) 20119861-119867 curve From Figure 11

AssemblyWorking air gap ℎ (mm) 01 012

ControlDriving current 119868 (A) 1sim18

working air gaps and drive currents are obtained by the forcesensor

A comparison between the simulated and experimentalresults of the electromagnetic force at various drive currentsandworking air gaps is shown in Figures 12 and 13 It is easy tosee that the simulated and experimental results match closelyat different drive currents for the two working air gaps andthe maximum error is 15 which could meet the demandsof the engineering applications Conclusively our developed

5 10 15 20 25 30 35 40 45 50 55 600Magnetic field intensity (H Am)

002040608

112141618

222

Mag

netic

indu

ctio

n in

tens

ity (B

T)

Figure 11 119861-119867 curve for the armature and iron core

numerical electromagnetic model of the HSV can predict theelectromagnetic force with acceptable accuracy

4 Results and Discussions

As shown in Figure 14 the electromagnetic force of the HSVincreaseswith the increasing drive current at working air gapsof 01mm and 012mm and it is larger with a working airgap of 01mm than with a working air gap of 012mm at thesame drive current This is because the electromagnetic forceof the HSV is inversely proportional to the reluctance undera constant drive current and coil turns based on (1)ndash(4) and(9) At the same time the smaller the working air gap is thesmaller the air gap reluctance is Hence the electromagneticforce at a working air gap of 01mm is larger than that ata working air gap of 012mm at the same drive currentThe influence of the drive current on the electromagneticforce is more complex On the one hand the drive currentaffects the total magnetic flux of the HSV and on the other


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)

Simulated resultExperimental data

0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ N

)

Figure 12 Comparison between simulated and experimental resultsof electromagnetic force at ℎ = 01mm

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ

N)


hand it also affects the transient magnetic permeability ofsoftmagnetic materials and thereby the reluctance of the ironcore and armatureTherefore to deeply analyze the influencemechanism of the drive current on the electromagneticenergy conversion of the HSV the relationship between theincrease in the electromagnetic force and the drive current isobtained by processing the test data of Figure 14 as shown inFigure 15

It can be observed fromFigure 15 that with the increase inthe drive current the increment of the electromagnetic forceincreases at first and then decreases reaching a maximum atthe drive current of 4A When the drive current is less than4A the increment of the electromagnetic force at a workingair gap of 01mm is always larger than that for a working airgap of 012mmwhen the drive current is greater than 4A theincrement of the electromagnetic force is almost coincidentat different working air gaps and the electromagnetic forceslowly increases as the increment gradually decreases In factthe change in the electromagnetic force is affected by the drivecurrent and the total reluctance composed of the reluctances

Air gap of 010 mmAir gap of 012 mm

0

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

Elec

trom

agne

tic fo

rce(

FＧＡ

N)

Figure 14 Experimental results of electromagnetic force at differentworking air gaps


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

02468

101214161820222426

Elec

trom

agne

tic fo

rce i

ncre

men

t(Δ

FＧＡ

N)

Figure 15The electromagnetic force increment at different workingair gaps

of the air gap and the soft magnetic materials of the ironcore and armature The contributions of these two factorsto the electromagnetic force are different within the rangeof different drive currents Consequently the angle of thevariation law of the total reluctance with the drive current canbetter explain the variation law of Figure 15

It can be observed from Figure 16 that the total reluctanceincreases with the increasing drive current at different airgaps when the drive current is less than 4A However theincrease degree of the reluctance is relatively low and notenough to significantly reduce the electromagnetic force ofthe HSV In this case the increase in the drive current makesthe dominant contribution to the electromagnetic force ofthe HSV Accordingly the increment of the electromagneticforce rapidly increases with the increase in the drive currentat small drive current values (shown in Figure 15) Whenthe drive current is more than 4A the increase of the totalreluctance is significantly influenced by the increase of thedrive current and the total reluctance increases rapidly withthe increase in the drive current Although the increaseof the total magnetic flux with the increase of the drive



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

123456789

10

Tota

l mag

netic

relu

ctan

ce(R

ＮＩＮＦ1

（)

times106

Figure 16 The total magnetic reluctance as a function of drivecurrent at different working air gap

current is beneficial to increasing the electromagnetic forcethe contribution rate of the total reluctance to reduce theelectromagnetic force is a priority Eventually the incrementof the electromagnetic force gradually decreases

It can be observed from Figure 16 that the total reluctanceof the HSV is different at different working air gaps when thedrive current is less than 4A and the smaller the workingair gap is the smaller the total reluctance is Accordingly theelectromagnetic force increment at a small working air gapis more than that at a large working air gap within a certainrange of the drive current As the drive current continues toincrease the variation curve of the total reluctance is almostcoincident at different working air gaps That is to say thechange in the total reluctance is determined by the reluctanceof the soft magnetic material rather than that of the workingair gap (however it also explains that the influence of the softmagnetic material should be considered in the mathematicalmodel of the HSV) Hence changes in the electromagneticforce are mainly affected by the total reluctance of the HSVat the moment according to the variation law shown in Fig-ure 15That is to say when the drive current is relatively largethe electromagnetic force increment at different working airgaps has the same variation law and the electromagnetic forceincrement decreases gradually

5 Conclusions

(1) 119861-119867 curve mathematical model easily applied withhigh precision is proposed and validated by the com-parison of the predicted and experimental results Itcan accurately describe the nonlinear magnetizationand magnetic saturation phenomena in the wholemagnetization process of amagneticmaterial includ-ing the initial magnetization phase the magneticsaturation phase and the transition phase from theunsaturated regime to the saturated state

(2) Based on the electromagnetic transient coupling prin-ciple an electromagnetic mathematical model of theHSV is developed in Fortran language that takes

the magnetic saturation characteristics into accountand it is validated by experimental data of the staticelectromagnetic force

(3) Experimental research on the variation relationshipbetween the static electromagnetic force of the HSVand its drive current at different working air gaps iscarried out It is concluded that with the increasingdrive current the increment of the electromagneticforce first increases rapidly and then decreases atdifferent working air gaps and there exists a drivecurrent value that makes the increment of the elec-tromagnetic force achieve its maximum value

(4) Based on the established electromagnetic mathe-matical model of the HSV it is revealed that theelectromagnetic energy conversion characteristics ofthe HSV are affected by the drive current and thetotal reluctance and that these two influence factorswithin the scope of the different drive currents havedifferent contribution rates to the electromagneticenergy conversion efficiency

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This work is supported by the Natural Science Founda-tion of Heilongjiang Province of China (LC201422) theNational Natural Science Foundation of China (NSFC51279037 51379041 51475100 and 51679048) the Fun-damental Research Funds for the Central Universities(HEUCF160304) and Recruitment Program of High-EndForeign Experts of the State Administration of ForeignExperts Affairs (GDW20162300256)

References

[1] L Su X Li Z Zhang and F Liu ldquoNumerical analysis onthe combustion and emission characteristics of forced swirlcombustion system for di diesel enginesrdquo Energy Conversionand Management vol 86 pp 20ndash27 2014

[2] L Su X Li X He and F Liu ldquoExperimental research onthe diffusion flame formation and combustion performance offorced swirl combustion system for DI diesel enginesrdquo EnergyConversion and Management vol 106 pp 826ndash834 2015

[3] X Li Z Sun W Du and R Wei ldquoResearch and developmentof Double Swirl combustion system for a di diesel enginerdquoCombustion Science and Technology vol 182 no 8 pp 1029ndash1049 2010

[4] X R Li H Q Zhou L W Su Y L Chen Z Y Qiao and F SLiu ldquoCombustion and emission characteristics of a lateral swirlcombustion system for ID diesel engines under low excess airratio conditionsrdquo Fuel vol 184 pp 672ndash680 2016

[5] F Luo H Cui and S Dong ldquoTransient measuring method forinjection rate of each nozzle hole based on spray momentumfluxrdquo Fuel vol 125 pp 20ndash29 2014


[6] Z Y Sun G X Li Y S Yu S C Gao andG X Gao ldquoNumericalinvestigation on transient flow and cavitation characteristicwithin nozzle during the oil drainage process for a high-pressure common-rail DI diesel enginerdquo Energy Conversion andManagement vol 98 pp 507ndash517 2015

[7] Z-Y Sun G-X Li C Chen Y-S Yu and G-X Gao ldquoNumer-ical investigation on effects of nozzlersquos geometric parameterson the flow and the cavitation characteristics within injectorrsquosnozzle for a high-pressure common-rail di diesel enginerdquoEnergy Conversion and Management vol 89 pp 843ndash861 2015

[8] Y Bai L Y Fan X Z Ma H L Peng and E Z Song ldquoEffectof injector parameters on the injection quantity of commonrail injection system for diesel enginesrdquo International Journal ofAutomotive Technology vol 17 no 4 pp 567ndash579 2016

[9] Q Dong W Long T Ishima and H Kawashima ldquoSpraycharacteristics of V-type intersecting hole nozzles for dieselenginesrdquo Fuel vol 104 pp 500ndash507 2013

[10] P Liu L Fan Q Hayat D Xu X Ma and E Song ldquoResearchon Key Factors andTheir Interaction Effects of ElectromagneticForce of High-Speed Solenoid Valverdquo Scientific World Journalvol 2014 Article ID 567242 2014

[11] Z-Y Sun G-X Li L Wang W-H Wang Q-X Gao and JWang ldquoEffects of structure parameters on the static electro-magnetic characteristics of solenoid valve for an electronic unitpumprdquo Energy Conversion and Management vol 113 pp 119ndash130 2016

[12] Q Cheng Z D Zhang H Guo and N L Xie ldquoImprovedprocessing and performance of GDI injector based on metalinjection molding technologyrdquo International Journal of AppliedElectromagnetics and Mechanics vol 44 no 1 pp 99ndash114 2014

[13] J I Miller T J Flack and D Cebon ldquoModeling the magneticperformance of a fast pneumatic brake actuatorrdquo Journal ofDynamic SystemsMeasurement and Control Transactions of theASME vol 136 no 2 Article ID 21022 2014

[14] Y Shin S Lee C Choi and J Kim ldquoShape optimization tominimize the response time of direct-acting solenoid valverdquoJournal of Magnetics vol 20 no 2 pp 193ndash200 2015

[15] G M Bianchi S Falfari F Brusiani et al ldquoAdvanced modellingof a new diesel fast solenoid injector and comparison withexperimentsrdquo SAE Technical Papers 2004

[16] K F Elmer and C R Gentle ldquoA parsimonious model for theproportional control valverdquo Proceedings of the Institution ofMechanical Engineers Part C Journal ofMechanical EngineeringScience vol 215 no 11 pp 1357ndash1363 2001

[17] J Ma G G Zhu and H Schock ldquoA dynamic model ofan electropneumatic valve actuator for internal combustionenginesrdquo Journal of Dynamic Systems Measurement and Con-trol Transactions of the ASME vol 132 no 2 pp 1ndash10 2010

[18] A H Shamdani A H Shamekhi M Ziabasharhagh and CAghanajafi ldquoModeling and simulation of a diesel engine com-mon rail injector in MatlabSimulinkrdquo in Proceedings of 14thAnnual International Conference of Mechanical EngineeringIsfahan Iran May 2006

[19] GM Bianchi P Pelloni F Filicori and G Vannini ldquoOptimiza-tion of the solenoid valve behavior in common-rail injectionsystemsrdquo SAE Technical Papers 2000

[20] Y Wang T Megli M Haghgooie K S Peterson and A GStefanopoulou ldquoModeling and control of electromechanicalvalve actuatorrdquo SAE Technical Papers 2002

[21] S K Chung C R Koch and A F Lynch ldquoFlatness-basedfeedback control of an automotive solenoid valverdquo IEEE Trans-actions onControl SystemsTechnology vol 15 no 2 pp 394ndash4012007

[22] B Huber and H Ulbrich ldquoModeling and experimental valida-tion of the solenoid valve of a common rail diesel injectorrdquo SAETechnical Papers vol 1 2014

[23] N-H Chung B-G Oh and M-H Sunwoo ldquoModellingand injection rate estimation of common-rail injectors fordirect-injection diesel enginesrdquo Proceedings of the Institution ofMechanical Engineers Part D Journal of Automobile Engineer-ing vol 222 no 6 pp 1089ndash1101 2008

[24] E E Topcu I Yuksel and Z Kamis ldquoDevelopment of electro-pneumatic fast switching valve and investigation of its charac-teristicsrdquoMechatronics vol 16 no 6 pp 365ndash378 2006

[25] Y Jin R Deng Y Jin and X Hu ldquoResearch on the responsecharacteristics of solenoid valve of the air-jet loom by simu-lationrdquo Journal of Thermal Science vol 22 no 6 pp 606ndash6122013

[26] P Naseradinmousavi and C Nataraj ldquoNonlinear mathematicalmodeling of butterfly valves driven by solenoid actuatorsrdquoApplied Mathematical Modelling vol 35 no 5 pp 2324ndash23352011

[27] A Mehmood S Laghrouche and M El Bagdouri ldquoNonlineardynamic modeling of an electro-pneumatic pressure converterfor VGT pneumatic actuatorrdquo International Journal of Automo-tive Technology vol 14 no 6 pp 941ndash953 2013

[28] G Sefkat ldquoThe design optimization of the electromechanicalactuatorrdquo Structural andMultidisciplinary Optimization vol 37no 6 pp 635ndash644 2009

[29] QWang F Yang Q Yang J Chen andHGuan ldquoExperimentalanalysis of new high-speed powerful digital solenoid valvesrdquoEnergy Conversion and Management vol 52 no 5 pp 2309ndash2313 2011

[30] M Coppo C Dongiovanni and C Negri ldquoNumerical analysisand experimental investigation of a common rail-type dieselinjectorrdquo Journal of Engineering for Gas Turbines and Power vol126 no 4 pp 874ndash885 2004

[31] H LiuHGu andDChen ldquoApplication of high-speed solenoidvalve to the semi-active control of landing gearrdquoChinese Journalof Aeronautics vol 21 no 3 pp 232ndash240 2008

[32] D T Vu Y Choi and J Kim ldquoLumped parametermodeling andanalysis of hybrid magnet engine valve actuatorrdquo InternationalJournal of Precision Engineering and Manufacturing vol 11 no6 pp 983ndash986 2010

[33] D C Jiles and D L Atherton ldquoTheory of ferromagnetichysteresis (invited)rdquo Journal of Applied Physics vol 55 no 6pp 2115ndash2120 1984

[34] J V Leite S L Avila N J Batistela et al ldquoReal coded geneticalgorithm for Jiles-Atherton model parameters identificationrdquoIEEE Transactions on Magnetics vol 40 no 2 pp 888ndash8912004

[35] M F Jaafar ldquoMagnetic hysteresis modeling and numericalsimulation for ferromagnetic materialsrdquo in Proceedings of 2013International Conference on Control Decision and InformationTechnologies CoDIT 2013 pp 516ndash523 tun May 2013

[36] J H Chan A Vladimirescu X C Gao P Liebmann and JValainis ldquoNonlinear transformer model for circuit simulationrdquoIEEE Transactions on Computer-Aided Design of IntegratedCircuits and Systems vol 10 no 4 pp 476ndash482 1991

[37] LMS Simens ldquoUserrsquos Guides of AMESim Rev12rdquo 2013

Submit your manuscripts athttpswwwhindawicom

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

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


structure parameters and the interaction between them onthe electromagnetic force using a combination of responsesurface design and FEM Sun et al [11] studied the influencerules of the iron core length the cross-sectional areas of themain and side poles the coil turns and the air gap widthon the electromagnetic force of an E-shaped HSV for anelectronic unit pump by FEM It was discovered that the drivecurrent significantly impacts the electromagnetic energyconversion as increasing the drive current unnecessarilydoes not improve the electromagnetic force of the HSV butonly increases the power consumption and thus reducesthe electromagnetic energy conversion efficiency Cheng etal [12] studied using FEM the flux density distribution ofan HSV whose iron core is made of a nano-based softmagneticmaterial It was discovered that solenoid valves withdifferent softmagneticmaterials have different characteristicsof electromagnetic force Miller et al [13] Shin et al [14]and Bianchi et al [15] carried out optimization research onthe static electromagnetic force of an HSV to obtain theoptimal structure parameters by FEM However because ofthe long solution time and high requirements for computerperformance of FEM the efficiency of research on theoptimization and analysis of the structural parameters is lowespecially when carrying out an iterative structural analysisfor the detailed parameter design of HSV Therefore thedevelopment of a static electromagnetic mathematical modelfor an HSV that can be applied conveniently with highaccuracy has become an important research direction

Elmer and Gentle [16] put forward a relatively simplemathematical model of a proportional solenoid valve Themodel simplified the electromagnetic process calculation ofthe solenoid valve as the calculation of an RL equivalentcircuit Ma et al [17] Shamdani et al [18] and Bianchi et al[19] determined the influencing relation of drive current andair gap on electromagnetic force by a polynomial fit techniquebased on test data of the static electromagnetic force Wanget al [20] and Chung et al [21] considered the magnetic sat-uration characteristics of an HSV by adjusting the coefficientof the fitting formula to limit the maximum electromagneticforce of the valve Huber and Ulbrich [22] and Chung etal [23] developed a dynamic characteristic mathematicalmodel for the common rail injector Its electromagneticsubmodel was built based on a map chart consisting of theelectromagnetic force drive current and air gap HowevertheHSV electromagnetic characteristics are not only a simplefunction relationship between the electromagnetic force andthe drive current and air gap In the high-pressure commonrail system the nonlinear transient coupling characteristicsof the electromagnetic mechanical and hydraulic systemsdetermine the electromagnetic characteristic transformationof the HSV Therefore this method meets the demands ofengineering application but it cannot reveal the intrinsicelectromagnetic properties of the HSV both the static elec-tromagnetic characteristics and the dynamic electromagneticcharacteristics

Most of the literature on the HSV tends to ignorethe influence of the soft magnetic material reluctance indeveloping the electromagnetic model of the solenoid valveTopcu et al [24] Jin et al [25] Naseradinmousavi and

Nataraj [26] and Mehmood et al [27] considered that thereluctance of the iron core magnetic material is much lessthan that of the air gap therefore the nonlinear mag-netization process and magnetic saturation characteristicsof the iron core magnetic material can be ignored Sefkat[28] assumed that the magnetic field is not saturated inthe whole working process of the solenoid valve and themagnetic induction intensity always linearly increases withthe increasing magnetic field intensity that is to say thesaturation phenomenon of the electromagnetic force did nothappen for their model However Wang et al [29] and Sunet al [11] discovered that a saturation phenomenon of theelectromagnetic force for the solenoid valve existed in theirexperimental research and the magnetic saturation of theHSV soft magnetic materials was the main cause of this Inthe actual control process of the HSV to improve the openingspeed a high voltagewas loaded on theHSV to quickly obtaina large current In this way it is easy to achieve magneticsaturation for the solenoid valve When the solenoid valvereached magnetic saturation with the increase in the drivecurrent the electromagnetic force of the HSV will increasevery slowly therefore the economy of the HSV decreasedSimply dealing with the magnetic saturation characteristic ortaking no account of it will lead to obvious calculation errorsin determining the electromagnetic force characteristicsTherefore it is necessary to take the nonlinear magnetizationcharacteristics and magnetic saturation characteristics of thesolenoid valvemagneticmaterials into account upon buildinga mathematical model of an HSV

Coppo et al [30] established a complete mathematicalmodel for a common rail injector For its electromagneticsubmodels the magnetization process of the solenoid valvewas divided into a linear magnetization phase and a satu-ration magnetization phase which are described by simplepiecewise lines In the mathematical model of an HSVdeveloped by Liu et al [31] the reluctance of the softmagneticmaterials was included to consider the nonlinear magnetiza-tion characteristic but the most important 119861-119867magnetizingcurve was determined based on 119861-119867 experimental data by atrial and error method for calculating the reluctance of softmagnetic materials The mathematical models of a solenoidvalve developed by Jin et al [25] and Vu et al [32] consideredthe magnetic saturation characteristics edge effect of theair gap and flux leakage but a detailed description of the119861-119867 magnetizing curve that determines the reluctance ofthe soft magnetic materials and calculates their transientpermeability was not provided

It can be found from the above references that a certainresearch insufficiency on the electromagnetic mathematicalmodel of the HSV still exists Some electromagnetic mathe-matical models use an electromagnetic force fitting formulabased on experimental data some models do not considerthe reluctance of the soft magnetic materials and someonly took segmental magnetic properties of the magneticmaterials into account Therefore to provide more valuableinformation on the electromagnetic energy conversion ofthe HSV and more deeply understand the solenoid valversquoselectromagnetic conversion process more detailed researchon the electromagnetic mathematical model of the HSV


Coil Coil

Armature

Iron core

r1

r2

r3

r4

r5

r6

r8r7

la

lb

lc

SＣＨ SＩＯＮ

ld

h






Φ = 119873 sdot 119868119877total (1)





N middot I


RＣＬＩＨ

Φ



119877gap1 = ℎ1205830 sdot 119878in (3)

119877gap2 = ℎ1205830 sdot 119878out (4)


119897119889120583 sdot 11987810158401015840 (5)


119897119886120583 sdot 1198781015840 (6)


119897119887 = 1199031 + 11990322

119897119886 = 1199035 + 1199036 minus 1199033 minus 11990344 119897119889 = 119897119887

11987811 = 120587 sdot (1199031 minus 1199032) sdot 1199035 + 1199036 + 1199033 + 11990344

11987822 = 120587 sdot 1199038 sdot 1199035 + 1199036 + 1199033 + 11990344

119878in = 1205874 sdot (1199032

4minus 11990323)

119878out = 1205874 sdot (1199032

6minus 11990325)

(7)



119865mag = 1212060121205830 sdot 119878in +

1212060121205830 sdot 119878out

(8)




119861+ = 119861119904 sdot 119867 + 1198671198881003816100381610038161003816119867 + 1198671198881003816100381610038161003816 + 119867119888 sdot (119861119904119861119903 minus 1)


119861 = 119861+ + 119861minus2

(9)


119861 = 1205830 (119867 +119872119904 sdot tanh(119867119860)) (10)

119860 = 119872119904(119861119904 (1205830 sdot 119867119904)) minus 1 (11)

0

05

1

15

2

25


Mag

netic

indu

ctio

n in

tens

ity (B

T)




Experimental data

002040608

112141618

222

Mag

netic

indu

ctio

n in

tens

ity (B

T)








Bs

Mag

netic

indu

ctio

n in

tens

ity (B

T)





119861 = ln (119867 + 1) (12)




004081216

224283236

Mag

netic

indu

ctio

n in

tens

ity (B

T)




004081216

224283236

444

Mag

netic

indu

ctio

n in

tens

ity (B

T)




119861 = 1199011 sdot ln (119867 + 1) (13)


119861 = ln (1199012 sdot 119867 + 1) (14)




0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)





0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)




119861 = 1199011 sdot ln (1199012 sdot 119867 + 1) (15)









Bench fixed end


Current probe

Driving system

Bench free end

Armature

Iron core

Force sensor



Currentprobe


Amplifier

Force sensor

Driv

ing

syste

m

(b) Test apparatus












002040608

112141618

222

Mag

netic

indu

ctio

n in

tens

ity (B

T)






0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ N

)


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ

N)





0

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

Elec

trom

agne

tic fo

rce(

FＧＡ

N)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

02468

101214161820222426

Elec

trom

agne

tic fo

rce i

ncre

men

t(Δ

FＧＡ

N)






1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

123456789

10

Tota

l mag

netic

relu

ctan

ce(R

ＮＩＮＦ1

（)

times106




5 Conclusions








Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Coil Coil

Armature

Iron core

r1

r2

r3

r4

r5

r6

r8r7

la

lb

lc

SＣＨ SＩＯＮ

ld

h






Φ = 119873 sdot 119868119877total (1)





N middot I


RＣＬＩＨ

Φ



119877gap1 = ℎ1205830 sdot 119878in (3)

119877gap2 = ℎ1205830 sdot 119878out (4)


119897119889120583 sdot 11987810158401015840 (5)


119897119886120583 sdot 1198781015840 (6)


119897119887 = 1199031 + 11990322

119897119886 = 1199035 + 1199036 minus 1199033 minus 11990344 119897119889 = 119897119887

11987811 = 120587 sdot (1199031 minus 1199032) sdot 1199035 + 1199036 + 1199033 + 11990344

11987822 = 120587 sdot 1199038 sdot 1199035 + 1199036 + 1199033 + 11990344

119878in = 1205874 sdot (1199032

4minus 11990323)

119878out = 1205874 sdot (1199032

6minus 11990325)

(7)



119865mag = 1212060121205830 sdot 119878in +

1212060121205830 sdot 119878out

(8)




119861+ = 119861119904 sdot 119867 + 1198671198881003816100381610038161003816119867 + 1198671198881003816100381610038161003816 + 119867119888 sdot (119861119904119861119903 minus 1)


119861 = 119861+ + 119861minus2

(9)


119861 = 1205830 (119867 +119872119904 sdot tanh(119867119860)) (10)

119860 = 119872119904(119861119904 (1205830 sdot 119867119904)) minus 1 (11)

0

05

1

15

2

25


Mag

netic

indu

ctio

n in

tens

ity (B

T)




Experimental data

002040608

112141618

222

Mag

netic

indu

ctio

n in

tens

ity (B

T)








Bs

Mag

netic

indu

ctio

n in

tens

ity (B

T)





119861 = ln (119867 + 1) (12)




004081216

224283236

Mag

netic

indu

ctio

n in

tens

ity (B

T)




004081216

224283236

444

Mag

netic

indu

ctio

n in

tens

ity (B

T)




119861 = 1199011 sdot ln (119867 + 1) (13)


119861 = ln (1199012 sdot 119867 + 1) (14)




0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)





0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)




119861 = 1199011 sdot ln (1199012 sdot 119867 + 1) (15)









Bench fixed end


Current probe

Driving system

Bench free end

Armature

Iron core

Force sensor



Currentprobe


Amplifier

Force sensor

Driv

ing

syste

m

(b) Test apparatus












002040608

112141618

222

Mag

netic

indu

ctio

n in

tens

ity (B

T)






0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ N

)


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ

N)





0

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

Elec

trom

agne

tic fo

rce(

FＧＡ

N)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

02468

101214161820222426

Elec

trom

agne

tic fo

rce i

ncre

men

t(Δ

FＧＡ

N)






1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

123456789

10

Tota

l mag

netic

relu

ctan

ce(R

ＮＩＮＦ1

（)

times106




5 Conclusions








Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119865mag = 1212060121205830 sdot 119878in +

1212060121205830 sdot 119878out

(8)




119861+ = 119861119904 sdot 119867 + 1198671198881003816100381610038161003816119867 + 1198671198881003816100381610038161003816 + 119867119888 sdot (119861119904119861119903 minus 1)


119861 = 119861+ + 119861minus2

(9)


119861 = 1205830 (119867 +119872119904 sdot tanh(119867119860)) (10)

119860 = 119872119904(119861119904 (1205830 sdot 119867119904)) minus 1 (11)

0

05

1

15

2

25


Mag

netic

indu

ctio

n in

tens

ity (B

T)




Experimental data

002040608

112141618

222

Mag

netic

indu

ctio

n in

tens

ity (B

T)








Bs

Mag

netic

indu

ctio

n in

tens

ity (B

T)





119861 = ln (119867 + 1) (12)




004081216

224283236

Mag

netic

indu

ctio

n in

tens

ity (B

T)




004081216

224283236

444

Mag

netic

indu

ctio

n in

tens

ity (B

T)




119861 = 1199011 sdot ln (119867 + 1) (13)


119861 = ln (1199012 sdot 119867 + 1) (14)




0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)





0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)




119861 = 1199011 sdot ln (1199012 sdot 119867 + 1) (15)









Bench fixed end


Current probe

Driving system

Bench free end

Armature

Iron core

Force sensor



Currentprobe


Amplifier

Force sensor

Driv

ing

syste

m

(b) Test apparatus












002040608

112141618

222

Mag

netic

indu

ctio

n in

tens

ity (B

T)






0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ N

)


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ

N)





0

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

Elec

trom

agne

tic fo

rce(

FＧＡ

N)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

02468

101214161820222426

Elec

trom

agne

tic fo

rce i

ncre

men

t(Δ

FＧＡ

N)






1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

123456789

10

Tota

l mag

netic

relu

ctan

ce(R

ＮＩＮＦ1

（)

times106




5 Conclusions








Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Bs

Mag

netic

indu

ctio

n in

tens

ity (B

T)





119861 = ln (119867 + 1) (12)




004081216

224283236

Mag

netic

indu

ctio

n in

tens

ity (B

T)




004081216

224283236

444

Mag

netic

indu

ctio

n in

tens

ity (B

T)




119861 = 1199011 sdot ln (119867 + 1) (13)


119861 = ln (1199012 sdot 119867 + 1) (14)




0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)





0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)




119861 = 1199011 sdot ln (1199012 sdot 119867 + 1) (15)









Bench fixed end


Current probe

Driving system

Bench free end

Armature

Iron core

Force sensor



Currentprobe


Amplifier

Force sensor

Driv

ing

syste

m

(b) Test apparatus












002040608

112141618

222

Mag

netic

indu

ctio

n in

tens

ity (B

T)






0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ N

)


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ

N)





0

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

Elec

trom

agne

tic fo

rce(

FＧＡ

N)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

02468

101214161820222426

Elec

trom

agne

tic fo

rce i

ncre

men

t(Δ

FＧＡ

N)






1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

123456789

10

Tota

l mag

netic

relu

ctan

ce(R

ＮＩＮＦ1

（)

times106




5 Conclusions








Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)





0

05

1

15

2

25

Mag

netic

indu

ctio

n in

tens

ity (B

T)




119861 = 1199011 sdot ln (1199012 sdot 119867 + 1) (15)









Bench fixed end


Current probe

Driving system

Bench free end

Armature

Iron core

Force sensor



Currentprobe


Amplifier

Force sensor

Driv

ing

syste

m

(b) Test apparatus












002040608

112141618

222

Mag

netic

indu

ctio

n in

tens

ity (B

T)






0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ N

)


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ

N)





0

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

Elec

trom

agne

tic fo

rce(

FＧＡ

N)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

02468

101214161820222426

Elec

trom

agne

tic fo

rce i

ncre

men

t(Δ

FＧＡ

N)






1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

123456789

10

Tota

l mag

netic

relu

ctan

ce(R

ＮＩＮＦ1

（)

times106




5 Conclusions








Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Bench fixed end


Current probe

Driving system

Bench free end

Armature

Iron core

Force sensor



Currentprobe


Amplifier

Force sensor

Driv

ing

syste

m

(b) Test apparatus












002040608

112141618

222

Mag

netic

indu

ctio

n in

tens

ity (B

T)






0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ N

)


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ

N)





0

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

Elec

trom

agne

tic fo

rce(

FＧＡ

N)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

02468

101214161820222426

Elec

trom

agne

tic fo

rce i

ncre

men

t(Δ

FＧＡ

N)






1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

123456789

10

Tota

l mag

netic

relu

ctan

ce(R

ＮＩＮＦ1

（)

times106




5 Conclusions








Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ N

)


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Drive current (I A)


0

20

40

60

80

100

120

140

160

Elec

trom

agne

tic fo

rce(

FＧＡ

N)





0

20

40

60

80

100

120

140

160

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

Elec

trom

agne

tic fo

rce(

FＧＡ

N)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

02468

101214161820222426

Elec

trom

agne

tic fo

rce i

ncre

men

t(Δ

FＧＡ

N)






1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

123456789

10

Tota

l mag

netic

relu

ctan

ce(R

ＮＩＮＦ1

（)

times106




5 Conclusions








Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 180Drive current (I A)

123456789

10

Tota

l mag

netic

relu

ctan

ce(R

ＮＩＮＦ1

（)

times106




5 Conclusions








Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




investigation on electromagnetic models of high-speed...

Documents