proc imeche part i: j systems and control engineering

Original Article

Proc IMechE Part I:J Systems and Control Engineering1–16� IMechE 2020Article reuse guidelines:sagepub.com/journals-permissionsDOI: 10.1177/0959651820979086journals.sagepub.com/home/pii

Identification of critical movingcharacteristics in high speed on/offvalve based on time derivative of thecoil current

Qiang Gao , Yuchuan Zhu , Changwen Wu and Yulei Jiang

AbstractThis article focuses on accurate identification of the critical moving characteristics of the high speed on/off valve.Typically, there are two strategies for identifying the critical moving characteristics, including calculation strategy basedon force balance and strategy for detecting the coil current’s certain points. However, the accuracy of the two abovestrategies needs to be improved. Therefore, to improve the identification accuracy of the high speed on/off valve’s criticalmoving characteristics, an identification strategy for detecting the time derivative of the coil current is proposed. First, amathematical model of the high speed on/off valve (including electromagnetic sub-model and mechanical-fluid sub-model)is established. And on this basis, relationship between the coil current’s derivative and the valve’s critical moving charac-teristics is analyzed which reveals the changing rule of the coil current’s derivative causing by the ball valve’s moving.Finally, the changing rule of the coil current’s derivative is verified by the comparative simulations and experiments, whichalso indicate that, with the proposed identification strategy, the maximum identification error of the critical opening/clos-ing time is only within 6%, and the maximum identification error of the total opening/closing time is still small (2.9%),compared to other identification strategies in the previous literatures.

KeywordsHigh speed on/off valve, critical moving characteristics, identification, coil current, derivative

Date received: 26 February 2020; accepted: 7 November 2020

Introduction

High speed on/off valve (HSV) are widely used in digi-tal hydraulic parallel system,1 antilock braking system(ABS),2 and fuel injectors3 due to some advantages,such as high reliability, high efficiency, and low cost,4,5

compared to the proportional/servo valve. From theperspective of the electro-mechanical converter, sole-noid actuator, magnetostrictive actuator,6 and piezo-electric actuator7 are all applied in the HSV. However,compared to the latter two actuators, HSV driven bysolenoid actuator has more significant advantages suchas larger displacement, smaller size, and lower cost.

The performance of the HSV driven by the solenoidactuator mainly consists of static flow rate and dynamicperformance. The former is depending on the ratio offull opening time and full closed time in one period ofpulse-width modulation (PWM) signal;8 the latterdepends on the total opening/closing time which affectsthe switching frequency of the HSV. As the total open-ing/closing time of the HSV decreases, the range of the

static flow rate and the switching frequency bothincrease, which can improve the output performance ofthe HSV. In addition, the total opening/closing time ofthe HSV are seriously affected by some nonlinear fac-tors such as the fluctuation of the hydraulic pressure,steady/transient flow force, and friction force.9

During the designed stage, the output displacementof the HSV can be directly measured by a laser sensorto determine the critical moving characteristics, includ-ing opening delay time, total opening time, closingdelay time, and total closing time. However, in manypractical applications, HSV is inserted in a valve body

National Key Laboratory of Science and Technology on Helicopter

Transmission, Nanjing University of Aeronautics and Astronautics,

Nanjing, China

Corresponding author:

Yuchuan Zhu, National Key Laboratory of Science and Technology on

Helicopter Transmission, Nanjing University of Aeronautics and

Astronautics, Nanjing 210016, China.

Email: [email protected]

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to control the flow rate or pressure of the system whichleads to impossibility of direct measurement of thevalve’s position. Therefore, the indirect measurementof HSV’s moving characteristics seems to be an effec-tive way.

Nowadays, there are two indirect measurementmethods of the HSV’s moving characteristics, includingposition estimation and coil current’s characteristicidentification.

With respect to position estimation, many identifica-tion methods considering noise are used to estimate theparameters of the nonlinear systems such as interval-valued type-2 robust fuzzy c-regression model, fuzzy c-regression, and possibilistic c-regression model.10,11

Rahman et al.12 use the solenoid’s incremental induc-tance to estimate the position of the valve. The incre-mental inductance is calculated by the rate of thecurrent rise of the PWM signal. In Zhao et al.,13 a non-linear sliding-mode observer (SMO) is proposed toobserve the spool position based on the voltage andcurrent values of the solenoid coil. In Nagai andKawamura,14 the relationship between the inductanceand the plunger position is measured in advance andthe position can be estimated based on the inductancewhich is calculated by the coil current. In addition, adisturbance observer is designed to control the positionof the plunger. Zhang et al.15 calculate the criticalopening current and critical closing current based onthe force balance of the ball valve, and the openingdelay time and closing delay time can be obtained insuccession. As can be seen from the previous publica-tions, the position estimation strategy is always used toachieve proportion control of the valve’s position, andits estimated accuracy depends on the accuracy of themeasured parameters and the model. However, due tothe nonlinear parameters and external disturbances, theestimated error of the valve’s position actually exists.

Compared to the position estimation, coil current’scharacteristic identification is a more effective and sim-ple way to determine the opening/closing characteristicsof the HSV. For example, Ahn and Yokota16 determinecritical opening time (the time when the valve starts tomove) and the total opening time (the time when thevalve reaches the fully open position) by a convex pointand an inflection point of the coil current curve in theopening stage. In this research, the two points aredetected by zooming the curve of the coil current. In Leet al.,17 some critical moving characteristics (openingdelay time, total opening time, closing delay time, andtotal closing time) of the pneumatic HSV are deter-mined based on the detection of the coil current. InXiong and Huang,18 a convex point on the coil currentcurve is used to represent the time when the valve startto move. However, there is a deficiency of the aboveresearch works that convex point and concave point ofthe coil current cannot represent the critical openingand closing time but only represent that the derivativeof the current is 0. This is indirectly verified by experi-ments in Yudell and Van de Ven.19 Therefore, the

critical opening and closing time of the HSV cannot beidentified by the coil current.

In this article, an identification strategy for detectingthe time derivative of the coil current is proposed toimprove the identification accuracy of the HSV’s criti-cal moving characteristics. First, a mathematical modelof the HSV (electromagnetic and mechanical-fluid sub-models) is established. Next, relationship between thecoil current and the ball valve’s critical moving charac-teristics is analyzed which can reveal the changing ruleof the coil current causing by the ball valve’s moving.In addition, the changing rule of the coil current’s deri-vative is verified by the comparative simulations andexperiments. The experimental results indicate that theproposed identification strategy is easy to implementand can accurately identify the critical moving charac-teristics of the HSV with smaller error, compared tothe previous studies.

Working principle and mathematicalmodel

A two-way and three-position HSV is used as the studyobject, as shown in Figure 1.

As shown in Figure 1, initially, the ball valve movesto the left side due to the hydraulic pressure in Port Pwhen the coil is de-energized, which means that thePort P connects to Port C; conversely, the ball valvemoves to the right side due to the electromagnetic forcewhen the coil is energized, which means that the Port Cconnects to Port T.

Electrical model

The balanced equation of coil voltage is written as

U(t)= I(t)R+dc(t)

dt= I(t)R+L(t)

dI(t)

dt+ I(t)

dL(t)

dt

ð1Þ

where U and I represent the driving voltage and coilcurrent; c represents the flux linkage; L and R repre-sent the equivalent inductance and resistance, respec-tively; In equation (1), R is assumed to be constant.

Figure 1. Schematic diagram of the HSV.

2 Proc IMechE Part I: J Systems and Control Engineering 00(0)

Electromagnetic model

Magnetic circuits are shown in Figure 2. Here, magneticcircuit 1 represents main flux path in working air gap,magnetic circuit 2 represents edge flux path of air gap,magnetic circuit 3 represents edge flux path in annular-ring portion, and magnetic circuit 4 represents leakageflux path in the coil’s radial direction;20 Rmy, Rag1, Rag2,Rp, Rar, and Rma are the reluctance of the magnet yoke,air gap 1, air gap 2, pole, armature, and main air gap,respectively.

Relationship between the magnetic flux and the totalreluctance can be defined as

NI(t)=f(t)Rto(t) ð2Þ

where N is the number of coil turns, f is the total mag-netic flux, and Rto is the total reluctance of magneticcircuit

Because the pole, armature, and magnet yoke arematerials with high permeability and the reluctance ofair gap is smaller than that of the main air gap, henceassuming that Rp, Rar, Rmy, Rag1, and Rag2 are ignored.The total reluctance Rto are written as

Rto(t)=Rma(t)=1

1R1(t)

+ 1R2(t)

+ 1R3(t)

+ 1R4

ð3Þ

where R1, R2, R3, and R4 are the reluctance of magneticcircuits 1, 2, 3, and 4, respectively.

Corresponding reluctances are defined as20

R1(t)=d(t)

m0S,S=p

D21

4, d(t)= d0 � x(t)

R2(t)=1

1:63m0(D1

2 + d(t)4 )

R3(t)=1

m0D1 ln (2D4�2D1

pd(t) )

R4 =1

m0ph8

D4 +D1

D4�D1� 1

2 (D4 +D1)+D2

4�D2

1

2ph

h i

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð4Þ

where d is the length of the main air gap; d0 is the initiallength of the main air gap; x is the displacement of theball valve; S is the effective sectional area of the arma-ture; h is the height of the coil; D1 and D4 are the dia-meter of armature and inner diameter of the magnetyoke, respectively; and m0 is the air permeability.

The magnetic leakage factor l is defined as

l(t)=R1(t)

Rto(t)ð5Þ

The dynamic electromagnetic force Fdm is written as

Fdm(t)=B2(t)S

2l(t)m0

ð6Þ

where Fdm is the dynamic electromagnetic force and Bis the magnetic flux density.

Assuming that the magnetic field is uniformly dis-tributed and the assembling air gap is negligibly small,a magnetic circuit equation is defined as21,22

NI(t)=Hc(t)Lc +Hg(t)d(t) ð7Þ

where Hc and Hg are the equivalent magnetic fieldintensity in the core and the air gap and Lc is the lengthof the magnetic circuit inside the core.

The relationship between B and H is written as

Hc(t)=B(t)

mc(t)ð8Þ

Hg(t)=B(t)

m0

ð9Þ

where mc is the core permeability.So, equation (7) is re-written as

NI(t)=B(t)

mc(t)Lc +

B(t)

m0

d(t) ð10Þ

The model of the core permeability is defined as

mc(t)=mi 1� B2(t)

B2s

� �ð11Þ

where Bs is the magnetic saturation flux density and mi

is the initial core permeability.So, substituting the mc in equation (11) into equation

(10), it is easy to get

NI(t)=B(t)Lc

mi(1� B2(t)B2s)+ d(t)

m0

24

35 ð12Þ

Equation (12) can be simplified to

B3(t)+P1(t)B2(t)+P2(t)B(t)+P3(t)=0 ð13Þ

where P1, P2, and P3 are defined as

Figure 2. Equivalent magnetic circuit of the HSV.

Gao et al. 3

P1(t)=�NI(t)m0

d(t)

P2(t)=� B2s +

m0LcB2s

d(t)mi

� �

P3(t)=NI(t)m0B

2s

d(t)

8>>>>>>><>>>>>>>:

ð14Þ

Letting

T1(t)=P2(t)

3� P2

1(t)

9

T2(t)=P3(t)

2+

P31(t)

27� P1(t)P2(t)

6

8>><>>:

ð15Þ

Q1(t)= sign(T2(t))ffiffiffiffiffiffiffiffiffiffiffiffiffiT1(t)j j

p

Q2(t)= arccosT2(t)

Q31(t)

� �8><>: ð16Þ

The solution of B can be calculated as

B(t)=2Q1(t) cosp +Q2(t)

3

� �� P1(t)

3ð17Þ

According to equations (7)–(9), the following equa-tion is obtained

NI(t)=Hc(t)Lc +Hg(t)d(t)

=Hc(t) Lc +mr(t)(d0 � x(t))½ �ð18Þ

where mr is the relative permeability (mc/m0).The coil inductance L is defined as

L(t)=c(t)

I(t)=

Nf(t)

I(t)ð19Þ

According to equations (9), (18), and (19), the coilinductance L is re-written as

L(t)=N2mc(t)S

Lc +mr(t)(d0 � x(t))=

N2m0SLc

mr(t)+ d0 � x(t)

ð20Þ

Taking the derivative of equation (20) with time, wecan get

dL(t)

dt=

N2m0S

Lc

mr(t)+ d0 � x(t)

� �2 dx(t)

dt+

Lc

m2r (t)

dmr(t)

dt

� �

ð21Þ

Mechanical-Fluid model

Assuming that the transient flow force of the ball valveis negligibly small, the dynamic model of the ball valvecan be simply defined as

md2x(t)

dt2=Fdm(t)� Bm

dx(t)

dt� psAs � Fs ð22Þ

where m is the equivalent moving mass of the ball valve;Bm is the coefficient of the damping; ps and As are thesupply pressure and the area of the inlet port, respec-tively; Fs is the steady flow force.

The steady flow force can be written as15

Fs =2CvCdApDp cos u ð23Þ

where Cv and Cd represent the fluid velocity coefficientand flow coefficient, respectively; Ap and Dp represent theopening area across orifice and pressure difference acrossthe orifice, respectively; and u represents the flow angle.

Validation of the simulation model

The main equations used for simulation include equa-tions (1), (6), (11), (20), (22), and (23). Based on theseequations, a simulation model is established byMATLAB/Simulink, as shown in Figure 3, in whichthe numerical integration method is set as Runge–Kutta and the fixed step is set as 0.05ms.

Main parameters are shown in Table 1. Some para-meters (m0, N, D2, D3, d0, and h) are from Gao et al.,22

the parameter (Dp) is from Zhang et al.,15 R is measuredexcept for the parameters (Bs and mi).

In addition, to estimate the parameters Bs and mi,the nonlinear least square and the sum squared errorare selected as the optimization method and the costfunction, respectively. In addition, the driving voltage(experiment) and the coil current (experiment) are

Figure 3. Simulation model of the high speed on/off valve.


selected as input data and the output data, which areall from Figure 4.

The estimation iteration process is shown in Figure5. Using the estimated parameters Bs and mi fromFigure 5, comparisons between the simulations andexperimental results under different amplitudes and fre-quencies are shown in Figure 6.

As shown in Figure 6, comparing the simulationresults with the experimental data under differentamplitudes and frequencies, the parameter identifica-tion process increased the accuracy of the simulationmodel. The error between the simulation and experi-ments may be caused by residual magnetism, the vary-ing resistance may be caused by temperature rise, andthe used function of the core permeability which cannotbe guaranteed to be exactly the same as the actualmaterial. Therefore, the parameters Bs and mi are deter-mined to be 2T and 1.88 3 1024H/m (150m0).

Analysis and identification of the HSV’scritical moving characteristics

Relationship between the current and the criticalmoving characteristics in previous publications

Recently, the coil current is usually used to identify thecritical moving characteristics of the HSV. Figure 6shows the relationship between the current and the crit-ical moving characteristics in previous studies.15–18

Here, ‘‘tcon’’ denotes the critical opening time when theball valve transits from status off to status on, ‘‘tton’’denotes the total opening time when the ball valvereaches the fully open position, ‘‘tcoff’’ denotes the criti-cal closing time when the ball valve transits from statuson to status off, ‘‘ttoff’’ denotes the total closing timewhen the ball valve reaches the fully closed position,‘‘tH’’ denotes the excitation time of the high voltage, Tdenotes the time of one period, and ‘‘t’’ denotes theduty ratio of the PWM signal.

Figure 7(a) shows that a previous study15 assumesthat the critical opening current (Icon) and the criticalclosing current (Icoff) can be calculated based on theforce balance of the ball valve as shown in equations(24) and (25). In addition, the ball valve is ensured toreach the fully open position when the correspondingcurrent is slightly bigger than the critical opening cur-rent (Icon). The ball valve is ensured to reach the fullyclosed position when the current is slightly smaller thanthe critical closing current (Icoff)

Icon =N

Lon

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2lm0S(psAs � Fs)

pð24Þ

Icoff =N

Loff

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2lm0S(psAs +Fs)

pð25Þ

where Lon and Loff denote the inductance in the open-ing and closing stage, respectively.

Conversely, as shown in Figure 7(b), previous stud-ies16–18 assume that the convex point and concave pointon the current curve can represent the critical openingtime (tcon) and critical closing time (tcoff) of the ballvalve, respectively. The total opening time (tton) andthe total closing time (ttoff) can be identified by the twoinflection points on the coil current curve.

Relationship between the coil current derivative andthe critical moving characteristics

In order to verify the deficiencies of the previous publi-cations,16–18 the relationship between the coil currentand the valve’s moving is analyzed and verified in detailthrough theory and simulation, as follows:

1. Stage 1 (0 ; tcon): high voltage is used, but the ballvalve maintains the fully closed position (x=0).

When the rising edge of the control signal is trig-gered, the coil is driven by high voltage. The electro-magnetic force increases but still smaller than the

Table 1. Main parameters of HSV.

Name Unit Symbol Value

Mass of ball valve g m 13.5Air permeability H/m m0 4p 3 1027

Number of coil turns – N 900Resistance of the coil O R 9Initial length of main air gap mm d0 0.45Magnetic saturationflux density

T Bs 2

Initial core permeability H/m mi 1.88 3 1024

Diameter of the ball valve mm Dp 2.4Diameter of the armature mm D1 8Inner diameter of the yoke mm D4 26Height of the coil mm h 17Sectional area of armature mm2 S 45Opening area of orifice mm2 Ap 1.02Area of the inlet port mm2 As 1.37Length of coremagnetic circuit

mm Lc 30

Damping coefficient N/(m/s) Bm 0.6Fluid velocity coefficient – Cv 0.9Flow coefficient – Cd 0.65Flow angle � u 20

Figure 4. Experimental data used for parameter estimation.

Gao et al. 5

(a) (b)

(c)

Figure 5. Estimation iteration process of the parameters: (a) Bs, (b) mi, and (c) sum squared error (cost function).

(a) (b)

(c) (d)

Figure 6. Comparisons between the simulations and experiments under different amplitudes and frequencies: (a) amplitude is 24 Vand frequency is 20 Hz, (b) amplitude is 24 V and frequency is 10 Hz, (c) amplitude is 18 V and frequency is 20 Hz, and (d) amplitudeis 18 V and frequency is 10 Hz.


hydraulic pressure, so the ball valve maintains the fullyclosed position (x=0) and its velocity is 0. The deriva-tive of the current I1 is written as

dI1(t)

dt=

UH � I1(t)R� I1(t)dL1(t)dt

L1(t)ð26Þ

where I1 and L1 denote the coil current and inductanceduring stage 1 (0 ; tcon).

Equation (26) shows that, with the increase in the coilcurrent I1, its derivation gradually decreases which leadsto the rise of the current I1 with a decreasing slope.

2. Stage 2 (tcon ; tton): high voltage is used and theball valve starts to open.

The coil is driven by high voltage in this stage. Theelectromagnetic force increases and is bigger than thehydraulic pressure, so the ball valve starts to open(x . 0). According to equations (20) and (21), theinductance L2 and dL2/dt both increase which meansthat L2 is bigger than L1 and dL2/dt is bigger than dL1/dt. Therefore, the derivative of the coil current I2 canbe re-written as

dI2(t)

dt=


L2(t)\

dI1(t)

dtð27Þ

where I2 and L2 denote the coil current and inductanceduring stage 2 (tcon ; tton).

Equation (27) shows that when the ball valve startsto open which leads to a sudden decrease in dI2/dt.Therefore, this can be used as a criterion for judgingthe critical opening of the ball valve.

In addition, the following condition is given

dI2(t)

dt=

. 0,dL2(t)

dt\

UH � I2(t)R

I2

=0,dL2(t)

dt=

UH � I2(t)R

I2(t)

\ 0,dL2(t)

dt.

UH � I2(t)R

I2(t)

8>>>>>>>><>>>>>>>>:

ð28Þ

From equation (28), the sign of the dI2/dt changes inthis stage, which means that the current I2 first increasesand then decreases. Therefore, there is a convex point(dI2/dt=0) in the corresponding current curve.

When the ball valve starts to open (critical openingtime, tcon), its velocity is 0. Therefore, according toequation (21), it is easy to get

dL2(t)

dtcon\ 0\

UH � I2(t)R

I2(t)ð29Þ

Therefore, the time when the ball valve starts to open(critical opening time) is not equal to that when dI2/dtis 0 (convex point in the current curve), which provesthat there actually exists a deficiency in the detection ofthe critical opening time in the previous publications.16–18

3. Stage 3 (tton ; tH): high voltage is used and theball valve reaches the fully open position(x=xmax).

In Stage 3, the electromagnetic force still increasesand the ball valve reaches the fully open position (x=xmax). According to equations (20) and (21), it can beknown that the inductance L3 increases and dL3/dtdecreases instantaneously which means that L3 (. 0) isbigger than L2 and dL3/dt (\ 0) is smaller than dL2/dt.Therefore, the derivative of the current I3 in this stagecan be re-written as

dI3(t)

dt=


L3(t)

.UH � I3(t)R

L3(t). 0

ð30Þ

where I3 and L3 denote the coil current and inductanceduring Stage 3 (tton ; tH).

According to equation (30), it is easy to get that thesign of the current derivative dI3/dt abruptlychanges from negative (Stage 2) to positive and a corre-sponding inflection point appears in the correspondingcurrent curve. This can be used as a criterion for jud-ging that the ball valve just reaches the fully openposition.

(a)

(b)

Figure 7. Relation between current and critical movingcharacteristics in the previous studies: (a) strategy 1 mentionedin the previous study15 and (b) strategy 2 mentioned in previousstudies.16–18

Gao et al. 7

4. Stage 4 (tH ; tT): low voltage is used and the ballvalve maintains the fully open position (x=xmax).

In order to reduce the power consumption at thisstage, the low voltage (UL) is used to drive the coil whenthe ball valve maintains the fully open position (x=xmax). The derivation of the current I4 in this stage is asfollows

dI4(t)

dt=

UL � I4(t)R� I4(t)dL4(t)dt

L4(t)\

dI3(t)

dtð31Þ

where I4 and L4 denote the coil current and inductanceduring Stage 4 (tH ; tT).


dI4(t)

dt=

. 0,UL . I4(t)R

=0,UL = I4(t)R

\ 0,UL \ I4(t)R

8><>: ð32Þ

Since this stage does not involve the dynamic char-acteristics of the HSV, equation (32) is not analyzed indetail.

5. Stage 5 (tT ; tT + tcoff): zero voltage is used andthe valve maintains the fully open position(x=xmax).

When the falling edge of the control signal is trig-gered, the coil is driven by zero voltage in this stage.The electromagnetic force decreases but still bigger thanthe hydraulic pressure, so the ball valve maintains thefully open position (x=xmax). The current derivative iswritten as

dI5(t)

dt=

0� I5(t)R� I5(t)dL5(t)dt

L5(t)\ 0 ð33Þ

where I5 and L5 denote the coil current and inductanceduring Stage 5 (tT ; tT + tcoff).

According to equation (33), it is easy to get thatwhen the zero voltage is applied on the coil, the currentderivation dI5/dt suddenly decreases, and a reducedinflection point appears in the corresponding currentcurve.

6. Stage 6 (tT + tcoff ; tT + ttoff): zero voltage isused and the valve starts to close.

The coil is still driven by zero voltage during thisstage. The electromagnetic force decreases and smallerthan the hydraulic pressure, so the ball valve starts toclose. According to equations (20) and (21), the induc-tance L6 and dL3/dt both decrease instantaneouslywhich means that L6 is smaller than L5 and dL6/dt issmaller than dL5/dt. Therefore, the derivative of thecurrent I6 in this stage can be re-written as

dI6(t)

dt=


L6(t).

dI5(t)

dtð34Þ

where I6 and L6 denote the coil current and inductanceduring Stage 6 (tT + tcoff ; tT + ttoff).

Equation (34) shows that when the ball valve startsto close which leads to a sudden increase in dI6/dt.Therefore, this can be used as a criterion for judgingthe critical closing time of the ball valve.


dI6(t)

dt=

\ 0,dL6(t)

dt. � R

=0,dL6(t)

dt= � R

. 0,dL6(t)

dt\ � R

8>>>>>><>>>>>>:

ð35Þ

From equation (35), the sign of the dI6/dt changes inthis stage which caused that the current I6 firstdecreases and then increases. Therefore, there is a con-cave point (dI6/dt=0) in the corresponding currentcurve.

When the ball valve starts to close (critical closingtime, tcoff), its velocity is 0. Therefore, according toequation (21), it is easy to get

dL6(t)

dtcoff. 0. � R ð36Þ

Therefore, the time when the ball valve starts to close(critical closing time) is not equal to that when dI6/dt is0 (concave point in the current curve), which provesthat there actually exists a deficiency in the detection ofthe critical closing time in the previous publications.16–18

7. Stage 7 (tT + ttoff ; T): zero voltage is used andthe valve reaches the fully closed position (x=0).

dI7(t)

dt=


L7(t)\ 0\

dI6(t)

dtð37Þ

where I7 and L7 denote the coil current and inductanceduring Stage 6 (tT + tcoff ; tT + ttoff).

According to equation (37), it is easy to get thatthe sign of the current derivation dI7/dt abruptlychanges from positive (Stage 6) to negative and a corre-sponding inflection point appears in the correspondingcurrent curve. This can be used as a criterion for jud-ging that the ball valve just reaches the fully closedposition.

Therefore, according to the analysis of the aboveseven stages, it is easy to get

1. In the previous publication,15 the critical openingcurrent (Icon) and the critical closing current (Icoff)are calculated based on the force balance.However, the accuracy of this calculation dependson the mathematical model. Actually, because ofsome parameter uncertainties, such as change inresistance caused by temperature and change inthe air gap caused by impact, the accuracy of thismethod needs to be further improved. In addition,


the total opening time (tton) and the total closingcurrent (ttoff) are conservatively determined whichcausing a big error.

2. There actually exist deficiencies in publications16–18

that convex point and concave point on the coil cur-rent curve cannot represent the critical opening time(tcon) and critical closing time (tcoff), respectively.

3. Not only the coil current I but also its derivativedI/dt is closely related to the moving characteristicsof the ball valve. Therefore, the derivative of thecoil current dI/dt can be used to identify the criticalmoving characteristics, as shown in Figure 8.

Compared simulation results

To verify the effectiveness of the proposed controlscheme, the following three strategies are compared.

1. Strategy 1: this strategy was mentioned in the pre-vious publication,15 in which the critical openingcurrent (Icon) and the critical closing current (Icoff)can be calculated by equations (24) and (25) basedon the force balance of the ball valve. In addition,the ball valve is ensured to reach the fully openposition when the corresponding current is slightlybigger than the critical opening current (Icon). Theball valve is ensured to reach the fully closed posi-tion when the corresponding current is slightlysmaller than the critical closing current (Icoff).

2. Strategy 2: this strategy was mentioned in previouspublications,16–18 in which the critical opening time(tcon) and the critical closing time (tcoff) are bothidentified by the convex point and concave pointon the coil current curve. In addition, the totalopening time (tton) and the total closing time (ttoff)are both identified by the two inflection points onthe coil current curve.

3. Proposed strategy: this strategy is proposed in thisresearch in which the four critical moving charac-teristics (tcon, tton, tcoff, and ttoff) are all identifiedby dI/dt.

4. Standard: in simulation, since the ball valve’s dis-placement is obtained directly, so the ‘‘Standard’’means that the four critical moving characteristicscan be directly identified by the displacement curveof the ball valve.

In order to verify the effectiveness of the proposedidentification strategy, comparative simulations areimplemented under different supply pressures ps (4 and6MPa). Comparisons of critical opening characteristicsunder different supply pressures are shown in Figure 9.

As shown in Figure 9(a), the ball valve starts tomove at 1.65ms and reaches the fully open position at2.85ms; strategy 1 can identify the critical opening time(tcon, 1.6ms) with small error but not the total openingtime (tton, 3.3ms); conversely, strategy 2 can accuratelyidentify the total opening time (tton, 2.85ms) but not

(a) (b)

Figure 9. Comparisons of critical opening characteristics under different supply pressures: (a) 4 MPa and (b) 6 MPa.

Figure 8. Relationship between the current derivative andcritical moving characteristics in this research.

Gao et al. 9

the critical opening time (tcon, 2.35ms); when using theproposed strategy, the critical opening time tcon,1.65ms) and the total opening time (tton, 2.85ms) areboth accurately identified.

In addition, Figure 9(b) shows that, with the increasein the supply pressure, the resistance of the ball valveincreases which leads to the increase in the critical open-ing time (tcon) and total opening time (tton).

Comparisons of critical closing characteristics underdifferent supply pressures are shown in Figure 10.

In conclusion, comparative simulation results underdifferent supply pressures are shown in Table 2.

Results shown in Table 2 clearly indicate that thetotal opening time (tton, 3.41ms) and the total closingtime (ttoff, 16.81ms) identified by the strategy 1 areboth bigger than the standard value; the critical open-ing time (tcon, 2.45ms) and the critical closing time(tcoff, 14.85ms) identified by the strategy 2 are alsoslightly bigger than the standard value because of using

the convex point and concave point. Conversely, theproposed strategy can accurately the critical movingcharacteristics of the HSV under different supply pres-sures, compared to strategy 1 and strategy 2.

Comparative experimental results

To verify the effectiveness of the proposed identifica-tion strategy, comparative experiments are conductedon a test bench, as shown in Figure 11. The test benchmainly consists of a hydraulic power supply (rated pres-sure: 10MPa and rated flow rate: 10L/min), xPC targetcontroller (Master/slave computer and PCI 6251 con-trol card), a high-frequency current probe (ShenzhenZhiyong, CP8000, rising time \ 7ns), a high-frequencypressure sensor (Kunshan Shuanqiao CYG1401F,response frequency: 20 kHz), a voltage amplifier(NanTong Longyi LYB-5010), and an HSV. The high-frequency current probe is used to measure the coil

(a) (b)

(b)

Figure 10. Comparisons of critical closing characteristics under different supply pressures: (a) 4 MPa and (b) 6 MPa.

Table 2. Comparative simulation results of the different strategies under different supply pressures.

Supply pressure: 4 MPa

Strategy 1 Strategy 2 Proposed Standard

Value Error Value Error Value Error Value

tcon (ms) 1.6 # 3% 2.35 " 42.4% 1.65 0 1.65tton (ms) 3.3 " 15.8% 2.85 0 2.85 0 2.85tcoff (ms) 15.55 # 4.3% 16.85 " 3.7% 16.25 0 16.25ttoff (ms) 21.9 " 13.2% 19.35 0 19.35 0 19.35




tcon (ms) 1.7 # 8.1% 2.45 " 32.4% 1.85 0 1.85tton (ms) 3.5 " 14.8% 3.05 0 3.05 0 3.05tcoff (ms) 12.9 " 4.0% 13.45 " 8.5% 12.4 0 12.4ttoff (ms) 19.3 " 22.9% 15.7 0 15.7 0 15.7


current of the HSV; the high-frequency pressure sensoris used to measure the pressure of HSV’s port C; allsensors are powered by a linear DC power supplier(Chaoyang Power 4NIC-X24). The picture of the testsystem is shown in Figure 12.

In the condition of the experiment, double voltage isused to control the HSV in which a high voltage UH

(24V) and a low voltage UL (5V) are used in the open-ing stage (0–4ms) and fully open stage (4–10ms),respectively.

In experiment, since the ball valve’s displacementcannot be directly measured by the sensor, ‘‘Standard’’means that the pressure in chamber C of HSV is usedto determine the critical opening time and the criticalclosing time because the pressure in chamber C is sensi-tive to the moving of the ball valve, and the two inflec-tion points on the coil current’s curve are used to

determine the total opening time and the total closingtime, respectively.

A mean filter is used to eliminate the problem ofnoise amplification caused by the current derivative butsimultaneously causing a time lag (0.05ms) whichshould be subtracted when using the current derivative.

Experimental results of opening/closing characteristicsunder supply pressure (4MPa) are shown in Figure 13.

As shown in Figure 13(a), the ball valve starts tomove at 2.15ms and reaches the fully open position at3.1ms; strategy 1 can identify the critical opening time(tcon, 1.65ms) and the total opening time (tton, 3.45ms)with small error; conversely, strategy 2 can accuratelyidentify the total opening time (tton, 3.1ms) but not thecritical opening time (tcon, 2.75ms); when using the pro-posed strategy, the critical opening time (tcon, 2.15ms)and the total opening time (tton, 3.1ms) are both accu-rately identified.

Figure 13(b) shows that, compared to the standardvalue, the strategy 1 can identify the total closing time(ttoff, 29.8ms) with big error; conversely, strategy 2 canaccurately identify the total closing time (ttoff, 24.8ms)but not the critical closing time (tcoff, 21.15ms); whenusing the proposed strategy, the critical closing time(tcoff, 19.55ms) and the total closing time (ttoff,24.75ms) are both identified with small error. Delay inrise of the control pressure caused by the control cham-ber’s volume needs to be compensated.

Comparative experimental results of the differentstrategies under different supply pressures are shown inTable 3. Results shown in Table 3 show that the errorsof critical moving time between the simulation andexperimental results are acceptable. Moreover, strategy1 cannot accurately identify the critical moving charac-teristics (especially tcon, tton, and ttoff). This may becaused by the error of the mathematical model, the lagof chamber pressure relative to the valve’s displace-ment, the temperature increase in the oil,15 pressure

Figure 11. Schematic diagram of the HSV’s dynamic test.

Figure 12. Picture of the test system.

Gao et al. 11

fluctuation, and changing of the inductance. In addi-tion, the critical opening time (tcon) and the critical clos-ing time (tcoff) identified by the strategy 2 are slightlybigger than the standard value due to using the convexpoint and concave point on the coil current curve.Conversely, the proposed strategy can accurately iden-tify the critical moving characteristics of the HSV underdifferent supply pressures, compared to strategy 1 andstrategy 2. The maximum identification errors of thecritical opening/closing time and the total opening/clos-ing time are within 6% and 2.9%, respectively.

Discussion

Influences of the mean filter

To eliminate noise and thorns caused by the derivativeof the coil current, a mean filtering method is imple-mented. In the experiment, the mean value is computedover a running average window of one cycle of thespecified fundamental frequency. Here, Ts denotes theperiod of a running average window (0.2ms)

Mean(dI(t)

dt)=

1

Ts

Z t

t�Ts

dI(t)

dtdt=

I(t)� I(t� Ts)

Tsð38Þ

To analyze the mean filter method on the experimen-tal results, comparisons between the raw signal and thefiltered signal of dI/dt are shown in Figure 14.

As shown in Figure 14, compared to the raw signal,the filtered signal not only keeps the changing charac-teristics but also reduces the noise and thorns, which isbeneficial to identify the critical moving characteristicsof the HSV. The only problem is that the filter methodleads to a delay of the signal which should be compen-sated when identifying the critical moving characteris-tics of the HSV.

In addition, the simulation results are basically con-sistent with the filtered results (dI/dt) in one PWM sig-nal period. The error between the simulation resultsand the filtered results in the closing stage may becaused by the residual magnetization of the corematerial.

Figure 13. Comparative experiments of critical opening/closing characteristics under 4 MPa: (a) opening stage and (b) closing stage.

Table 3. Comparative experimental results of the different strategies under different supply pressures.




tcon (ms) 1.65 # 20.3% 2.75 " 27.9% 2.25 " 4.7% 2.15tton (ms) 3.45 " 11.3% 3.1 0 3.05 # 1.6% 3.1tcoff (ms) 14.8 # 21.9% 21.15 " 13.1% 19.55 " 4.5% 18.7ttoff (ms) 29.8 " 20.2% 24.8 0 24.75 # 0.2% 24.8




tcon (ms) 1.75 # 30% 3.05 " 22% 2.65 " 6% 2.5tton (ms) 3.55 " 4.4% 3.4 0 3.5 " 2.9% 3.4tcoff (ms) 14.55 " 2.1% 15 " 5.26% 14.4 " 1.1% 14.25ttoff (ms) 23.35 " 32.7% 17.6 0 17.75 " 0.9% 17.6


The only purpose of the filter is to clearly detect thelocal changing characteristic of the coil current deriva-tive, according to the above analysis, it is easy to knowthat the mean filter can meet the requirement of thisresearch and is easy to implement in actual application.

Influences of parameters (R and d0) on theidentification results

Since the coil resistance R is sensitive to the tempera-ture rise and the initial length of air gap d0 is affectedby the impact of the ball valve, it is necessary to analyzethe influences of these parameters’ variations on theidentification results of the critical moving time (tcon,tton, tcoff, and ttoff) of the HSV.

Influences of the coil resistance R on the identifica-tion results are shown in Figure 15.

As shown in Figure 15, the variations of R havesmall influences on the identification results of the criti-cal opening time (tcon) and the total opening time (tton);conversely, with the increase in R, the critical closingtime (tcoff) and the total closing time (ttoff) reduce sig-nificantly. For example, when the R increases from 8 to10O, the critical closing time (tcoff) reduces from 17.8to 13.4ms, and the total closing time (ttoff) reduces 21.9to 17.2ms, respectively.

This is because that when the voltage drops to 0,smaller R leads to the larger coil current which needsmore time to drop due to the effect of biggerinductance.

(a) (b)

(c)

Figure 14. Influences of filter on the comparisons between the simulations and experiments: (a) one period, (b) opening stage, and(c) closing stage.

(a) (b)

Figure 15. Influences of R on the identification of critical moving time: (a) opening stage and (b) closing stage.

Gao et al. 13

Influences of the initial length of air gap d0 on theidentification results are shown in Figure 16.

As shown in Figure 16, the variations of d0 almosthave no influence on the identification results of thecritical opening time (tcon) and the total opening time(tton); conversely, with the increase in the d0, the criticalclosing time (tcoff) and the total closing time (ttoff) bothreduce significantly. For example, when the d0 increasesfrom 0.43 to 0.47mm, the critical closing time (tcoff)reduces from 16.9 to 14ms, and the total closing time(ttoff) decreases from 20.9 to 17.9ms.

This is because that when the voltage drops to 0,smaller d0 leads to the larger magnetic field energywhich needs more time to naturally demagnetize.Therefore, a negative voltage is usually used in the clos-ing stage to accelerate the demagnetization.9,15,22

Conclusion

In this article, an identification strategy for detectingthe time derivative of the coil current is proposed toimprove the identification accuracy of the HSV’s criti-cal moving characteristics. Comparative theoreticalanalysis, simulations, and experiments have been car-ried out to illustrate the effectiveness of the proposedstrategy

1. Relationship between the coil current and the ballvalve’s critical moving characteristics is analyzedwhich reveals the changing rule of the coil currentand its derivative. Compared to the coil current, thederivative of the coil current is more sensitive to thecritical moving characteristics of the ball valve.Therefore, the derivative of the coil current can beused to identify the critical moving characteristics

2. In the previous publication,15 because of someparameter uncertainties, such as change in resis-tance caused by temperature and change in the air

gap caused by impact, the identification accuracyof critical moving characteristics needs to be fur-ther improved. In addition, the total opening time(tton) and the total closing current (ttoff) are conser-vatively determined which causing a big error.

3. It proves that the previous publications16–18 havemisunderstandings that the convex point and con-cave point on the coil current curve represent thecritical opening time and critical closing time of theball valve, respectively.

4. Compared to the strategies in previous studies, theproposed identification strategy is easy to imple-ment and can accurately identify the four criticalmoving characteristics with smaller error (6%)under different supply pressures.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research, authorship, and/or publi-cation of this article.

Funding

The author(s) disclosed receipt of the following finan-cial support for the research, authorship, and/or publi-cation of this article: This work was supported by theNational Natural Science Foundation of China (GrantNo. 51975275), National Key Laboratory of Scienceand Technology on Helicopter Transmission (NanjingUniversity of Aeronautics and Astronautics) (GrantNo. HTL-A-20G02), and Postgraduate Research &Practice Innovation Program of Jiangsu Province (No.KYCX20_0178).

ORCID iDs

Qiang Gao https://orcid.org/0000-0002-9318-4501Yuchuan Zhu https://orcid.org/0000-0002-7399-1656

(a) (b)

Figure 16. Influences of d0 on the identification of critical moving time: (a) opening stage and (b) closing stage.


https://orcid.org/0000-0002-9318-4501

https://orcid.org/0000-0002-7399-1656

https://orcid.org/0000-0002-7399-1656

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Appendix 1

Notation

Ap opening area across the orificeAs area of the inlet portB(t) magnetic flux densityBm damping coefficientBs magnetic flux saturation densityCd flow coefficientCv fluid velocity coefficientD1 diameter of armatureD4 inner diameter of the magnet yokeFdm(t) dynamic electromagnetic forceFs steady flow force.h height of the coilHc(t) equivalent magnetic field intensity in the

coreHg(t) equivalent magnetic field intensity in the

air gapI(t) coil currentIcoff critical closing currentIcon critical opening currentI1–7 coil current in defined stages 1–7L(t) inductanceLc length of the core magnetic circuitLoff inductance in closing stageLon inductance in opening stageL1–7 inductance in defined stages 1–7m equivalent moving mass of the ball valveN number of the coil turnsps supply pressureR equivalent resistanceRag1 reluctance of the air gap 1Rag2 reluctance of the air gap 2Rar reluctance of the armatureRma(t) reluctance of the main air gapRmy reluctance of the magnet yokeRp reluctance of the poleRto(t) total reluctance of the magnetic circuitR1(t) reluctance of the magnetic circuits 1R2(t) reluctance of the magnetic circuits 2R3(t) reluctance of the magnetic circuits 3R4 reluctance of the magnetic circuits 4

Gao et al. 15

S effective sectional area of the armaturetcoff critical closing timetcon critical opening timetH excitation time of the high voltagettoff total closing timetton total opening timeT period of the PWM signalU(t) driving voltageUH high voltage in opening stageUL low voltage in maximum opening stagex(t) displacement of the ball valved(t) length of the main air gap

d0 initial length of the main air gapDp pressure difference across the orificeu flow anglel(t) magnetic leakage factormc(t) core permeabilitymi initial core permeabilitymr(t) core relative permeabilitym0 air permeabilityt duty ratio of the PWM signalf(t) total magnetic fluxc(t) flux linkage


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