On/off solenoid valves with PWM control are widely used in all types of vehicle electro-hydraulic control systems respecting to their desirable properties of reliable, low cost and fast acting. However, it can hardly achieve a linear hydraulic modulation by using on/off valves mainly due to the nonlinear behaviors of valve dynamics and fluid, which affects the control accuracy significantly. In this paper, a linear relationship between limited pressure difference and coil current of an on/off valve in its critical closed state is proposed and illustrated, which has a great potential to be applied to improve hydraulic control performance. The hydraulic braking system of case study is modeled. The linear correspondence between limited pressure difference and coil current of the inlet valve is simulated and further verified experimentally. Based on validated simulation models, the impacts of key parameters are researched. The limited pressure difference affected by environmental temperatures is experimentally studied, and the amended linear relation is given according to the test data.

1. Research Article
Study on a linear relationship between limited pressure difference
and coil current of on/off valve and its influential factors
Junzhi Zhang n
, Chen Lv 1
, Xiaowei Yue 1
, Yutong Li 1
, Ye Yuan 1
State Key Laboratory of Automotive Safety and Energy, Department of Automotive Engineering, Tsinghua University, Beijing 100084,
People's Republic of China
a r t i c l e i n f o
Article history:
Received 17 May 2013
Received in revised form
10 August 2013
Accepted 6 September 2013
Available online 30 September 2013
This paper was recommended for
publication by Dr. Q.-G. Wang
Keywords:
Hydraulic modulation
Linear relationship
Pressure difference limiting
On/off valves
Influential parameters
a b s t r a c t
On/off solenoid valves with PWM control are widely used in all types of vehicle electro-hydraulic control
systems respecting to their desirable properties of reliable, low cost and fast acting. However, it can
hardly achieve a linear hydraulic modulation by using on/off valves mainly due to the nonlinear
behaviors of valve dynamics and fluid, which affects the control accuracy significantly. In this paper, a
linear relationship between limited pressure difference and coil current of an on/off valve in its critical
closed state is proposed and illustrated, which has a great potential to be applied to improve hydraulic
control performance. The hydraulic braking system of case study is modeled. The linear correspondence
between limited pressure difference and coil current of the inlet valve is simulated and further verified
experimentally. Based on validated simulation models, the impacts of key parameters are researched. The
limited pressure difference affected by environmental temperatures is experimentally studied, and the
amended linear relation is given according to the test data.
& 2013 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Hydraulic actuators have many desirable properties, such as
reliable, clean, low cost and fast acting, that makes them a natural
choice for various types of electro-hydraulic braking control systems
in passenger vehicle, including anti-lock braking system (ABS),
vehicle stability control (VSC), and regenerative braking system
(RBS) [1–4].
With development of automotive technology and increased
requirements on brake safety, comfort and high-efficiency,
improvement in precise and effective braking control is in need.
To achieve high-performance modulation of braking pressure,
proportional valve with digital control is the most effective and
direct way. It can achieve a continuous control of hydraulic fluid
flow, leading to a linear control of hydraulic pressure.
Although some of the proportional valves have been developed
with application of advanced nonlinear control techniques [5–8],
this kind of approach is usually highly cost and complicated with
hardware, which restricts their practical applications and on the
other hand makes on/off valves driven by pulse width modulated
(PWM) inputs widely used in vehicle's braking control system [9].
For example, the electrically-controlled braking system (ECB), which
has been employed in commercialized Toyota HEV Prius, is com-
prised of fourteen solenoid valves. Among the fourteen valves in use,
however, only two are proportional valves, while the others are all
on/off ones [10]. Similar situation can also be found in slip-control
boost (SCB) braking system developed by famous braking compo-
nent supplier TRW [11].
By utilizing on/off solenoid valves with PWM control, the costs
and complexity of the system do can be reduced effectively and
acceptable control accuracy can be obtained, but nonlinearity of
the modulated pressure is increased mainly due to the inherently
nonlinear discrete behavior of on/off valve and flow. Thus, to
improve the modulation performance, researchers worldwide have
been carried out comprehensive research in parameter design and
control method of hydraulic actuators.
Champagne and Stephens [12] carried out research on valve
actuator parameters optimizing to enhance the dynamic perfor-
mance of the control valve. The effects of supply pressure, step size,
load margin, flow, fluid volume and design style are investigated.
Muto and Yamada etc. [13] proposed a method for realizing the
smooth motion of the system by adjusting the phase difference
between the input signals to the valves. In the system to which this
method was applied, it was confirmed that the dynamic performance
of the hydraulic actuator was remarkably improved. Ahn and Yokota
[14] designed a modified PWM algorithm increasing the system
response of on/off valves, and proposed an on/off algorithm for
control parameters using a learning vector quantization neural net-
work, guaranteeing the effectiveness of the proposed control with
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2013.09.008
n
Corresponding author. Tel.: þ86 10 62771839.
E-mail addresses: jzhzhang@mail.tsinghua.edu.cn (J. Zhang),
henrylvchen@163.com (C. Lv).
1
Tel.: þ86 10 62771839.
ISA Transactions 53 (2014) 150–161

2. various external loads. Varseveld and Bone [15] used the typical two-
valve circuit and tried to improve the input–output linearity of the
whole system by changing the valve pulsing schemes, and they
obtained a quasi-linear input–output behavior. Actuating perfor-
mance of valves in a portable hardware-in-the-loop (HIL) device
for automotive diagnostic control systems was studied in [16]. Jeong
and Kim [17] studied the impacts of three major system variables,
namely the on- and the off-times of the valve and the system
configuration coefficient, on the mean pressure and the pressure
ripple amplitude under PWM control. Intelligent algorithm was
designed and proved to have good robustness and uncertainty
handling properties in identification and position control of servo
hydraulic rotary actuators including servo valves [18]. Wang and
Song etc. [19] studied a simulation model of an on/off valve in ABS/
ESP using PWM control with high modulation frequency at 2–4 kHz,
and discussed the quasi-proportional function of valve under differ-
ent duty cycles. The present authors modeled the dynamics of a
pneumatic valve, researching the PWM regulation of pneumatic
brake [20–22]; and studied the hydraulic pressure-increase rates
under varied PWM duties of on/off valves, developing a threshold
hydraulic pressure modulation algorithm for regenerative braking
control of an electric car [4,23]. Nevertheless, the existing research on
on/off valve control is mainly focusing on novel modulation method
and algorithm based on PWM control, which can achieve the quasi-
linear modulating performance, but a linear modulation effect has
rarely been seen.
In this paper, a linear relationship between limited pressure
difference and coil current of an on/off valve in its critical closed
state is proposed and researched, which has a great potential to be
applied to improve hydraulic control performance. The hydraulic
braking system of case study is modeled. The linear correspon-
dence between limited pressure difference and coil current of the
inlet valve is simulated and further verified experimentally. Based
on validated simulation models, the impacts of key parameters of
valve on the limited pressure difference are researched. And the
linear relationship proposed affected by environmental tempera-
tures is also experimentally studied. According to the test data, the
amended linear relation is given. Finally, some concluding remarks
are provided.
2. Linear relationship between limited pressure difference and
coil current of on/off valve
Take the inlet valve of the hydraulic braking system, a typical
normally opened outflow valve shown in Fig. 1, as an example.
Regarding the endpoint of valve core in the valve-closed state as the
origin, the OX coordinate system was established, as Fig. 2 shows.
OX is the movement direction of valve core, pointing forwards.
In valve closed state, the axial balance equation of valve core
can be expressed as:
ÀFe þFs þFh þFN ¼ 0 ð1Þ
where Fe is the electromagnetic force, Fs is the spring force, Fh is
the hydraulic force, and FN is the supportive force.
However, when valve reaches a critical state, i.e. it is still closed
(xv ¼ 0), but it's just about to open, the supportive force will
disappear (FN ¼ 0), thus Eq. (1) can be expressed as:
ÀFe þFs þFh ¼ 0 ð2Þ
The electromagnetic force acting on valve core is mainly deter-
mined by coil current I, turn number N, the air gap length l, and the
magnetic reluctance of air gap Rg. The amount of electromagnetic
force can be given by the relation:
Fe ¼ ðI Â NÞ2
=ð2Rg  lÞ ð3Þ
Linearizing Eq. (3), the electromagnetic force can be repre-
sented as follows:
Fe ¼
∂Fe
∂i
IðtÞþ
∂Fe
∂xe
xvðtÞ
¼ Ki  IðtÞþKxe  xvðtÞ ð4Þ
Under critical balanced state, the valve opening is xv ¼ 0, thus:
Fe ¼ Ki  IðtÞ ð5Þ
where Ki is the current–force gain, and Kxe is the displacement–
force gain.
For a normally opened valve, in the coordinate system defined,
the spring force can be given by the relation:
Fs ¼ Ks  ðx0 þxm ÀxvÞ ð6Þ
When reaching the valve critical balanced state (xv ¼ 0),
Fs ¼ Ks  ðx0 þxmÞ ð7Þ
where x0 is the pretension displacement of return spring, xm is the
maximum displacement of return spring, xv is the displacement of
return spring, and Ks is the stiffness coefficient.Fig. 1. Configuration of the inlet valve in hydraulic braking system.
Fig. 2. Diagram of the coordinate system of inlet valve.
J. Zhang et al. / ISA Transactions 53 (2014) 150–161 151

3. Under valve critical closed state, the hydraulic force acting on
valve core, which is caused by the hydrostatic pressure, can be
expressed as:
Fh_st ¼ πR2
v ð cos αÞ2
 Δp ð8Þ
where α is the cone angle of valve seat, Rv is the sphere radius of
valve core, Δp is the pressure drop between valve inlet and outlet.
Thus, Eq. (2) can be represented as:
ÀKi  IþKs  ðx0 þxmÞþπR2
v ð cos αÞ2
 Δp ¼ 0 ð9Þ
According to the equation above, a linear correspondence
between the pressure difference and the coil current under valve
critical balance state can be obtained, as Eq. (10) shows.
Δp ¼
Ki
πR2
v ð cos αÞ2
 IKs  ðx0 þxmÞ
πR2
v ð cos αÞ2
ð10Þ
The calculation and test results of the corresponding relation
between limited pressure difference and coil current of on/off
valve can be seen from Fig. 3.
Therefore, if the input pressure can be detected, a linear
characteristic of the expected output pressure can be obtained
based on the limited pressure difference, as Eq. (11) shows.
pout ¼ pin ÀΔp
¼ pin À
Ki
πR2
v ð cos αÞ2
 Iþ
Ks  ðx0 þxmÞ
πR2
v ð cos αÞ2
ð11Þ
As shown in Fig. 4, assuming that one operating point that
makes valve reaching the critical closed state is ðI0; Δp0Þ, if the
inlet pressure continues to be increased with coil current
unchanged, the pressure difference will become greater, leading
to the critical balance state broken, and the valve will open with
the operating point changing to ðI0; Δp0 þΔp′Þ. After flow com-
mences, as the inlet pressure decreases and outlet pressure
increases, the pressure difference will become smaller gradually.
Once the pressure difference recovers to Δp0, the critical balance
state of valve core will be regained. The valve will close again and
the working point will return to ðI0; Δp0Þ. On the other side, if the
inlet pressure is reduced, the pressure difference will decrease
accordingly. And close state of valve will continue to be kept with
the output pressure remaining unchanged.
In this way, the valve can be modulated in a “pressure difference
limiting” (PDL) mode, operating in its critical closed state; the limited
pressure difference between valve inlet and outlet is linearly
correlated with coil current. The linear characteristic described above
has a great potential to be utilized to improve control precision and
accuracy of hydraulic modulation.
3. Modeling and simulation
Although the correlation between pressure difference and coil
current presents a simple linearity, further study on its character-
istics including dynamic modulating performance via model-
based simulation and analysis are needed.
3.1. System modeling
In order to simulate and analyze the characteristics of linearity
between limited pressure difference and coil current, appropriate
dynamic models of the hydraulic system including valve actuators
need to be built up.
The ABS hydraulic modulator is a typical hydraulic control
system of automobile. Take the inlet valve of an ABS modulator as
a case to study, and the schematic diagram of the hydraulic system
selected is shown in Fig. 5. The inlet valve is set in between master
cylinder and wheel cylinder. pm is the master cylinder pressure,
which can be regarded as the inlet pressure of inlet valve, and pw is
the wheel cylinder pressure, which can be seen as the outlet
pressure of inlet valve. The structure of the wheel cylinder is
simplified to a piston and a spring.
3.1.1. Valve dynamics
As shown in Fig. 6, in the OX coordinate system defined, the
axial dynamic balance equation of valve core can be expressed as.
m Â
d
2
xv
d
2
t
¼ ÀFe þFs þFh ÀFB ð12Þ
where m is the mass of valve core and plug, xv is the displacement
of the valve core, and FB is the viscous force.
The electromagnetic force and the spring force are analyzed in
Section 2.Fig. 3. Correspondence between limited pressure difference and coil current.
Fig. 4. Three operating states of an on/off valve.
Fig. 5. Diagram of the hydraulic ABS braking system.
J. Zhang et al. / ISA Transactions 53 (2014) 150–161152

4. For viscous force, which is a resistance exerting on the core
when flow commences, it is affected by the viscosity of the fluid
and the movement velocity of valve core, as shown in Eq. (13).
FB ¼ B Â
dxv
dt
ð13Þ
where B is the viscous damping coefficient.
And after flow starts, the axial hydraulic force acting on the
core will become the hydrodynamic force, other than the hydro-
static one. The hydrodynamic force can be divided into two parts:
the stationary part and the transient part [24], shown in Eq. (14).
Fh ¼ Fst þFtrans ð14Þ
The stationary part of hydrodynamic force is caused by the non
time-varying flow. According to momentum theory, it can be
calculated as follows:
Fst ¼ πR2
v ð cos αÞ2
 ΔpÀ2C2
dΔp cos α Â Aj ð15Þ
Aj ¼
πdm
Rv
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2
v À
d
2
m
4
s
 xv ð16Þ
where Aj is flow area, dm is the average diameter of valve seat, Rv is
the sphere radius of valve core.
The transient hydrodynamic force, which is caused by time-
varying flow and related to the opening xv, can be calculated by
relation:
Ftrans ¼ ÀρL
dQ
dt
ð17Þ
where L is the damping length, ρ is the density of the hydraulic
fluid, qv is the hydraulic fluid flow of valve.
Thus, the hydrodynamic force can be given by the relation:
Fh ¼ πR2
v ð cos αÞ2
 ΔpÀ2C2
dΔp cos α
Â
πdm
Rv
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2
v À
d
2
m
4
s
xv ÀρL
dQ
dt
ð18Þ
3.1.2. Wheel cylinder pressure
The wheel cylinder pressure can be regarded as the outlet
pressure of inlet valve, which is expected to be obtained as
demand. The schematic diagram of the wheel cylinder is shown
in Fig. 5. The hydraulic fluid flow of a valve can be expressed as.
Q ¼ CdAj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 Â Δp
ρ
s
ð19Þ
where Cd is the flow coefficient of the inlet of the valve, taken
as 0.7.
Δp can be calculated by the following equations:
Δp ¼ pm Àpw ð20Þ
for the wheel cylinder
Awdx ¼ Qdt ð21Þ
dpw ¼
kw
Aw
dx ð22Þ
where kw is the spring stiffness of the wheel cylinder, Aw is the
cross-sectional area of wheel cylinder.
Combining Eqs. (21) and (22), pw can be represented as
dpw
dt
¼
kw
A2
w
CdAj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 Â Δp
ρ
s
ð23Þ
3.2. Verification of simulation model
First, using system parameters of Table 1, two scenarios,
namely: scenario 1, varied inlet pressure (6 MPa, 4 MPa, 2 MPa)
under the same coil current (150 mA); and scenario 2, same inlet
pressure (4 MPa) under varied coil current (150 mA, 175 mA,
200 mA), are simulated based on system dynamic models built
in Section 3. The two scenarios described above indicate two
typical braking processes of vehicle respectively: varied braking
demand of driver and different expected braking requests of
vehicle. Simulation results, shown in Fig. 7, verify the linearity of
PDL modulation proposed and correspondence between limited
pressure difference and coil current.
To validate the feasibility and accuracy of the simulation
models built above, experimental tests are carried out on an ABS
test bench with the same scenario set up of simulations.
The hardware-in-loop test bench is comprised of an upper
computer, a lower computer, an electric control unit, hydraulic
braking actuators and sensors. The upper computer is utilized to
monitor the testing process, analyze and save experimental
results. The lower computer adopted is an AutoBox from dSPACE.
Vehicle system dynamic models are running in AutoBox, simulat-
ing vehicle's behavior under different operating conditions. The
electric control unit consisting of a microcontroller, some periph-
eral circuits and driving circuits for valves, is adopted as a real one,
Table 1
Key parameters of the inlet valve.
Parameter Value
Coil resistance (R/Ω) 7.4
Turn number (N) 450
Mass of valve core (m/g) 2.1
Sphere diameter (Dv/mm) 1.588
Spring stiffness (N/mm) 0.34
Max displacement of core (xm/mm) 0.3
Pretension displacement of spring (x0/mm) 3.1
Cone angle of valve seat (α/degree) 114
Big-endian diameter of valve seat (d2/mm) 1.2
Little-endian diameter of valve seat (d1/mm) 0.65
Fig. 6. Schematic diagram of the valve core and flow.
J. Zhang et al. / ISA Transactions 53 (2014) 150–161 153

5. and communicates with dSPACE via CAN Bus. The brake actuators
are real ones from a passenger car, including a brake pedal, a
master cylinder, four wheel cylinders, and a standard ABS hydraulic
modulator. Sensors can acquire hydraulic pressure signals of master
and wheel cylinders, and send them to the lower computer and
electric control unit. Therefore, a closed-loop experimental platform
is formed as Fig. 8 shows.
According to Fig. 9, comparisons between experimental and
simulation results demonstrate that the simulation can reflect the
response of the real braking system under PDL modulation
properly, validating the feasibility and effectiveness of the simula-
tion models built.
3.3. Characteristic analysis of pressure difference limiting
modulation
For the hydraulic braking system studied, there are only two
inputs, namely: inlet pressure (master cylinder pressure) and coil
current, which could exert profound impacts on modulation of the
pressure difference limiting proposed. Thus, based on simulation
models built and verified above, characteristics of the linearity
under varied inlet pressure and varied coil current are analyzed
respectively as follows.
Fig. 8. The HIL test bench for ABS braking control of vehicle.
Fig. 9. HIL test verification results of PDL modulation.
Fig. 7. Simulation results of PDL modulation.
J. Zhang et al. / ISA Transactions 53 (2014) 150–161154

6. 3.3.1. Characteristic of PDL under varied inlet pressure
Under a certain coil current (taking as 140 mA), making the
input pressure of valve input changing from 1.5 MPa to 4.5 MPa,
the simulation results are shown in Fig. 10(a). Under four different
inlet pressures, although the outlet pressures vary from 0.3 MPa to
3.3 MPa, respectively, each corresponding limited pressure differ-
ence finally levels off at the same value of 1.2 MPa. The final
pressure difference has no relation with the initial input pressure,
demonstrating the linear characteristic of PDL modulation ana-
lyzed in Section 2. However, it does take a longer time for the
pressures modulation before reaching the stable state under a
higher inlet pressure.
3.3.2. Characteristic of PDL under varied coil current
Under a certain inlet pressure (setting as 3 MPa), change the coil
current inputs from 130 mA to 190 mA. As shown in Fig. 10(b), with
the coil current's increase, the outlet pressure decrease from 2.1 MPa
to 0.7 MPa, leading to the corresponding limited pressure difference
increasing from 0.9 MPa to 2.3 MPa. The simulation results indicate
that under the pressure difference limiting control mode, a larger coil
current input results in a higher limited pressure difference.
4. Impacts of valve structure parameters on limited pressure
difference
Since structure parameters of valve exert great impacts on
dynamic performance of the hydraulic system, the effects of those
key valve parameters, including cone angle of valve seat, mass and
sphere diameter of valve core, and spring stiffness, on pressure
difference limiting are studied in this section. And some important
factors, including limited pressure difference, displacement of
valve core, and hydrodynamic force, which can represent dynamic
characteristic of hydraulic modulating process, are focused on.
4.1. Mass of valve core
Making the mass of valve core change from 1.05 g to 2.1 g and
4.2 g, with other parameters kept unchanged, simulations are carried
out respectively. According to the results shown in Fig. 11(a), the
change of valve core's mass exerts no effect on the final limited
pressure difference. However, it leads to different pressures modula-
tion cycle before valve reaches the balanced state. As valve core's mass
increases, the period of pressure modulating process becomes longer,
as Fig. 11(b) shows. It is because that the movement of valve core can
be described as a two-order oscillation system. The adjust time is
negatively correlated with natural undamped angular frequency
ts pð1=ωnÞ, and valve core's mass is negatively correlated with natural
undamped angular frequency ωn pð1=mÞ, therefore the increase of
valve core's mass makes the pressure modulation period longer.
4.2. Sphere diameter of valve core
To study the impact of valve core's sphere diameter on PDL
modulation, simulations are carried out with sphere diameter
changing from 1.588 mm to (1.588À0.2) mm and (1.588þ0.2) mm.
According to the simulation results shown in Fig. 12, as the sphere
diameter increases, the effect of pressure difference limiting declines,
resulting in the limited pressure difference decreasing, the valve fully
opened period prolonging, and the amplitude of hydrodynamic force
becoming greater. Based on the balance equation of valve core, in the
critical closed state, a larger sphere diameter produces greater
hydraulic force acting on valve core under the same pressure
difference, taking up a bigger part of the electromagnetic force. Thus,
the pressure difference limiting effect is reduced, and the pressure
adjusting process becomes severer.
4.3. Cone angle of valve seat
Making the cone angle of valve seat change from 901 to 1141 and
1251, while other parameters are kept unchanged, the simulations are
Fig. 10. (a) Variation of limited pressure difference under varied inlet pressure; (b) variation of limited pressure difference under varied coil current.
J. Zhang et al. / ISA Transactions 53 (2014) 150–161 155

7. carried out. Simulation results shown in Fig. 13(a) indicate that the
cone angles’ change exerts great impact both on limited pressure
difference and dynamic procedure of pressure modulation. Based on
the simulation results shown in Fig. 13(b), the greater cone angle is,
the shorter modulation process achieves, i.e. the system response
becomes faster. It is because that the amount of cone angle is
negatively correlated with valve flow, i.e. qv p cos À1
α. As the cone
angle increases, the valve flow becomes bigger, making the modula-
tion process shorter.
4.4. Spring stiffness
Simulations are carried out with the stiffness coefficient of
spring varying from 170 N/m to 340 N/m and 510 N/m with other
parameters remaining unchanged. Simulation results in Fig. 14
show that the increase of spring's stiffness coefficient would
decrease the pressure difference limiting effect and prolong the
period of valve fully opening. It is because that the growth of stiffness
coefficient makes spring force greater, counteracting a larger part of
Fig. 11. Simulation results of PDL modulation under different masses of valve core.
J. Zhang et al. / ISA Transactions 53 (2014) 150–161156

8. electromagnetic force, which leads to a reduction of the ability to
limit pressure difference of valve.
4.5. Summary of influential parameters
Simulation results above show that mass of valve core exert
tiny impact on the final limited pressure difference, but affect the
dynamic performance of hydraulic modulating process; and varia-
tions of sphere diameter of valve core, stiffness coefficient of
spring and cone angle of valve seat would influence the limited
pressure difference directly. Detailed simulation data is shown in
Table 2.
5. Impact of environmental temperature on limited pressure
difference
Apart from the key parameters of system, temperature condi-
tions can significantly affect the performances and even the overall
functionality of electro-mechanical system. Temperature affects
performances of actuators, seal efficiency, lubricant and fluid
Fig. 12. Simulation results of PDL modulation under different sphere diameter of valve core.
J. Zhang et al. / ISA Transactions 53 (2014) 150–161 157

9. properties, which may induce deteriorating control performance
and problems [25]. Therefore, research the impact of environ-
mental temperature on modulating performance of pressure
difference limiting is of great necessity and importance.
5.1. Limited pressure difference affected by environmental
temperature
To evaluate the control performance of pressure difference
limiting affected by temperature, HIL tests under varied environ-
mental temperatures (À3 1C, 2 1C, 3 1C, 5 1C, 13 1C and 24 1C) were
conducted.
According to test results shown in Fig. 15, under a certain
value of coil current, valve's ability to limit pressure difference
drops gradually as environmental temperature rises. Taking the
coil current of 150 mA as an example, the limited pressure
difference decreases from 3.18 MPa at À3 1C to 2.43 MPa at
24 1C, dropping more than 0.7 MPa in an environmental tempera-
ture range of 27 1C.
Fig. 16 shows the variation of corresponding line between coil
current and limited pressure difference under different environ-
mental temperatures. Based on the test results, the curve moves
up by nearly 50% as temperature decreases by 27 1C, but the linear
relationship between coil current and limited pressure difference
is still kept relatively well.
Fig. 13. Simulation results of PDL modulation under different cone angles of valve seat.
J. Zhang et al. / ISA Transactions 53 (2014) 150–161158

10. The main reason for the phenomena described above is that the
coil resistance is affected by environmental temperature. The
amount of coil resistance is related to temperature, which can be
expressed by Eq. (24).
R ¼ R0 þαTðα40Þ ð24Þ
where R0 is the reference value of coil resistance, α is temperature
coefficient, T is temperature.
When temperature reduces, the coil resistance will decrease
accordingly, resulting in the electromagnetic force becoming
greater, increasing valve's ability to limit pressure difference.
Therefore, the limited pressure difference rises. Detailed experi-
mental data are shown in Appendix A.
5.2. Amendment of linear relation between limited pressure
difference and coil current
Based on the test results analyzed above, the limited pressure
difference between valve inlet and outlet is affected by variation
of environmental temperature significantly. Thus, the linearity of
Fig. 14. Simulation results of PDL modulation under different stiffness coefficient of spring.
J. Zhang et al. / ISA Transactions 53 (2014) 150–161 159

11. limited pressure difference is amended according to the test
results. Taking the linear relation between limited pressure differ-
ence and coil current at normal room temperature (24 1C) as a
reference, which is described as Eq. (10), under a certain range of
environmental temperatures (À5 1C to 25 1C), the linear relation
can be amended as:
Δp ¼
Ki
πR2
v ð cos αÞ2
 IKs  ðx0 þxmÞ
πR2
v ð cos αÞ2
À0:02ðT À297Þ ð25Þ
where T is environmental temperature (degree Kelvin).
6. Conclusion
Respecting to desirable properties of reliable, low cost and fast
acting, on/off solenoid valves with PWM control are widely used in
all types of vehicle electro-hydraulic control systems, including
braking control system. However, it can hardly achieve a linear
hydraulic modulation by using on/off valves mainly due to the
nonlinear behaviors of valve dynamics and fluid, which affects the
control accuracy significantly.
In this paper, a linear relationship between limited pressure
difference and coil current of an on/off valve in its critical closed
state is proposed and researched, which has a great potential to be
applied to improve hydraulic control performance. The hydraulic
braking system of case study is modeled. The linear correspon-
dence between limited pressure difference and coil current of the
inlet valve is simulated and further verified experimentally. Based
on validated simulation models, the impacts of key parameters of
valve on the pressure difference limiting are researched. The
limited pressure difference affected by environmental tempera-
tures is experimentally studied, and the amended linear relation is
given based on experimental test data.
Further studies will be carried out in some areas such as the
following: application of the linear characteristic of on/off valve
PDL modulation to vehicle braking control, including ABS and
regenerative brake; life test of on/off valves under pressure
difference limiting modulation.
Acknowledgements
The authors would like to thank the Natural Science Founda-
tion of China [project no. 51075225] and National High Tech
Project “863” of China [project no. 2011AA11A243] for funding
this work.
Appendix A
Notation
Aj flow area of valve opening
Aw cross-sectional area of wheel cylinder
B viscous damping coefficient
Cd flow coefficient
dm average diameter of valve seat
d1 little-endian diameter of valve seat
d2 big-endian diameter of valve seat
Dv sphere diameter of valve core
Fe electromagnetic force acting on valve
Fs spring force acting on valve
Fh hydraulic force acting on valve
FN supportive force acting on valve
Fst stationary hydrodynamic force
Ftrans transient hydrodynamic force
FB viscous force acting on valve
I coil current
kw spring stiffness of the wheel cylinder
Ki current–force gain
Kxe displacement–force gain
Fig. 15. Experimental results of limited pressure difference affected by temperature.
Fig. 16. Linear correspondence between limited pressure difference and coil
current affected by different temperatures.
Table 2
Simulation data of PDL modulation affected by valve parameters.
Valve
parameters
Limited
pressure difference
(MPa)
Adjust time
(ms)
Amplitude
of hydrodyna force
(N)
Cone angle of valve seat α/degree
90 0.69 366 2.287
114 1.16 369 1.427
125 1.62 416 1.122
Valve core mass (m/g)
1.05 1.15 310 1.466
2.1 1.16 349 1.427
4.2 1.17 461 1.455
Sphere diameter of valve core (Dv/mm)
1.388 1.53 432 1.181
1.588 1.16 387 1.427
1.788 0.92 369 1.796
Spring stiffness (N/mm)
0.17 2.15 257 1.669
0.34 1.16 377 1.427
0.51 0.19 581 1.249
J. Zhang et al. / ISA Transactions 53 (2014) 150–161160

12. Ks stiffness coefficient of valve return spring
l air gap length
L damping length
m mass of valve core and plug
N turn number
Δp pressure difference between valve inlet and outlet
pm master cylinder pressure
pw wheel cylinder pressure
Q hydraulic fluid flow of valve
Rg magnetic reluctance of air gap
Rv sphere radius of valve core
x0 pretension displacement of valve return spring
xm maximum displacement of valve return spring
xv displacement of the valve core
α cone angle of valve seat
ρ density of the hydraulic fluid
Appendix B. The auxiliary results
See Appendix Table B1.
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Table B1
Experimental test data of limited pressure difference under different temperatures.
Coil Current (mA) Limited pressure difference (MPa)
24 1C 13 1C 5 1C 3 1C 2 1C À3 1C
130 1.07 1.22 1.35 1.45 1.52 1.60
133 1.10 1.32 1.45 1.52 1.55 1.67
137 1.15 1.38 1.52 1.58 1.60 1.73
140 1.23 1.40 1.57 1.62 1.65 1.80
144 1.35 1.53 1.60 1.70 1.72 1.90
147 1.40 1.65 1.72 1.82 1.85 2.01
150 1.53 1.70 1.85 1.92 1.98 2.10
154 1.60 1.85 1.91 1.98 2.10 2.18
158 1.68 1.92 2.02 2.09 2.20 2.25
161 1.75 2.05 2.12 2.25 2.32 2.35
165 1.85 2.15 2.22 2.32 2.40 2.48
168 1.92 2.22 2.33 2.37 2.53 2.57
172 2.05 2.32 2.40 2.48 2.61 2.70
175 2.15 2.38 2.46 2.56 2.65 2.78
179 2.22 2.43 2.57 2.68 2.70 2.83
182 2.32 2.58 2.65 2.75 2.79 2.88
186 2.38 2.65 2.76 2.80 2.83 2.90
189 2.43 2.73 2.83 2.90 2.95 3.02
193 2.58 2.80 2.91 3.01 3.08 3.17
196 2.65 2.88 2.93 3.13 3.13 3.20
200 2.73 2.93 3.03 3.20 3.23 3.28
203 2.80 3.10 3.10 3.28 3.33 3.38
207 2.88 3.18 3.23 3.33 3.43 3.45
210 2.93 3.23 3.32 3.48 3.55 3.55
214 2.97 3.35 3.43 3.55 3.63 3.65
217 3.02 3.41 3.57 3.60 3.73 3.83
221 3.13 3.43 3.65 3.75 3.83 3.95
J. Zhang et al. / ISA Transactions 53 (2014) 150–161 161