Department of Mechanics,
Mathematics and Management (DMMM),
Polytechnic University of Bari,
Via Orabona 4,
Bari 70125, Italy;
Centre for Power Transmission and
Motion Control (PTMC),
Department of Mechanical Engineering,
University of Bath,
Bath BA2 7AY, UK
Andrew R. Plummer
Centre for Power Transmission and
Motion Control (PTMC),
Department of Mechanical Engineering,
University of Bath,
Bath BA2 7AY, UK
Department of Mechanics,
Mathematics and Management (DMMM),
Polytechnic University of Bari,
Via Orabona 4,
Bari 70125, Italy
Department of Mechanics,
Mathematics and Management (DMMM),
Polytechnic University of Bari,
Via Orabona 4,
Bari 70125, Italy
A Review of Direct Drive
Spool Valves: Industrial
State-of-the-Art and Research
This paper reviews the state of the art of directly driven proportional directional hydrau-
lic spool valves, which are widely used hydraulic components in the industrial and trans-
portation sectors. First, the construction and performance of commercially available
units are discussed, together with simple models of the main characteristics. The review
of published research focuses on two key areas: investigations that analyze and optimize
valves from a ﬂuid dynamic point of view, and then studies on spool position control sys-
tems. Mathematical modeling is a very active area of research, including computational
ﬂuid dynamics (CFD) for spool geometry optimization, and dynamic spool actuation and
motion modeling to inform controller design. Drawbacks and advantages of new designs
and concepts are described in the paper. [DOI: 10.1115/1.4041063]
Keywords: proportional valves, direct-drive, ﬂow forces, discharge coefﬁcient, CFD,
Proportional valves are critical components in many hydraulic
actuation and power transmission systems. They are used where
ﬂow rate, and hence, actuation speed needs to be accurately con-
trolled. Typical applications include mobile hydraulics (excava-
tors, wheel loaders, etc.), machine tools, industrial automation,
and marine hydraulics. However, the terms “proportional valve,”
“servovalve,” and “direct-drive valve” are not well deﬁned and
sometimes used interchangeably. In this paper, we are concerned
with spool valves in which the spool is directly driven by an elec-
trical actuator, speciﬁcally a proportional solenoid. Spool position
both determines the direction of ﬂow and modulates the ﬂow rate.
In contrast, a servovalve spool is driven by a faster, more power-
ful and more linear actuator, typically a hydraulic pilot stage, and
is manufactured to ﬁner tolerances. Servovalves often have nomi-
nally zero overlap (dead band), whereas proportional valves are
designed to have appreciable overlap .
A review of servovalve technology and research can be found
in Ref. . A servovalve requires more precise manufacturing
tolerances than a proportional valve, and to achieve this, a servo-
valve is usually designed with the spool sliding in a bushing
sleeve made of the same material as that of the spool. Detailed
metering features in a servovalve can be obtained by providing
the sleeve with slots. Instead, the spool in a proportional valve
directly slides in the valve body, and notches and grooves are
machined on the spool to achieve the desired ﬂow rate trend ver-
sus spool position . The smaller overlap of servovalves is also
synonymous with better response speed. This characteristic is fur-
ther enhanced in servovalves by high speed spool actuation in
which the pilot stage serves as a hydraulic ampliﬁcation system
capable of generating high pressure differences across the end
faces of the main spool, which in consequence is moved by a very
high actuation force. Instead, proportional directional valves are
commonly moved directly by proportional solenoids, whose
actuation forces are lower than those obtained in servovalves .
Note, however, that to control higher ﬂow rates, multistage pro-
portional valves may be used, which employ a small pilot spool to
actuate a large main stage spool.
In addition to the lower response speed (which is also due to
the high moving masses of direct actuation systems), direct oper-
ated proportional directional valves are not capable of producing
such a high “chip shear force,” namely, the force necessary to
shear contamination particles that can be caught between the
edges of a metering section. These drawbacks make proportional
Contributed by the Dynamic Systems Division of ASME for publication in the
JOURNAL OF DYNAMIC SYSTEMS,MEASUREMENT,AND CONTROL. Manuscript received
February 26, 2018; ﬁnal manuscript received July 28, 2018; published online
October 5, 2018. Assoc. Editor: Heikki Handroos.
Journal of Dynamic Systems, Measurement, and Control FEBRUARY 2019, Vol. 141 / 020801-1
C2019 by ASME
valves unsuitable for critical applications, such as aerospace
(where the additional size and weight is also a problem). How-
ever, by virtue of their robustness and relatively low cost com-
pared to servovalves, proportional valves are extensively used in
many industrial applications. Unlike an on/off valve, a propor-
tional valve can be instrumental in avoiding sudden acceleration
and deceleration of an actuator, in addition to providing more
accurate control of its position and/or velocity .
Given their importance in several industrial sectors, this paper
discusses the state of the art of directly driven proportional direc-
tional valves. First, their operating principles and mathematical
models used by researchers and industrial engineers to study and
design these valves will be discussed. Then, an overview of com-
mercially available valves will be given, with the emphasis on
their performance. Finally, a detailed review of the current
research will be provided that is focused on the ﬂuid dynamic
analysis and on spool position control systems.
Operating Principles and Analytical Modeling
Directly driven proportional directional valves have an inner
sliding spool which is directly moved by either one solenoid or
two solenoids placed at the spool extremities. The spool is pro-
vided with notches and grooves designed to achieve a desired
ﬂow rate trend as a function of the spool position . These valves
usually present a dead band given by the spool overlap, which can
be as high as 10% or more of the spool stroke. In addition, usually
the spool moves in a bore directly drilled in the valve body .
These valves are used with hydraulic oils, although some attempts
have been made in the scientiﬁc literature to effectively adapt
these valves to water [5–9].
Figure 1shows a typical architecture of the most used valve
typology, namely a four-way three-position (4/3) proportional
valve along with its symbol. The sliding spool is pushed directly
by either the right solenoid or the left solenoid depending on the
required hydraulic connections (P-A and B-T or P-B and A-T)
. The oil enters the valve through the high pressure port P,
then it ﬂows through the metering section P-A or P-B (whose ﬂow
area is determined by both the metering notches in the spool and
the opening degree) and ﬁnally exits the valve toward the actuator.
Likewise, the oil discharged from the actuator re-enters the valve
ﬂowing through the metering section A-T
. Ports T
are internally connected (not represented in Fig. 1for simplic-
ity) so as to form a single discharge port T .
Analytical models were developed in the past [12,13] and are
currently used in scientiﬁc literature to easily study these valves
[14–18]. The ﬂow rate through a proportional directional valve
depends on the opening area and on the pressure drop through the
valve. If Dpis the pressure drop measured across a metering edge
(x) the metering section area (a function of the spool posi-
tion x), the volumetric ﬂow rate Qthrough the metering chamber
can be calculated as
where Cdis the discharge coefﬁcient of the metering section. The
desired function A
(x) is obtained by properly designing the
notches machined on the spool.
In the case of a 4/3 valve, a discharge coefﬁcient must be
deﬁned for each of the two metering edges in the ﬂow path. In
such a case, the overall discharge coefﬁcient through the valve
can be calculated as proposed in Ref. 
where Cd;1and Cd;2are the discharge coefﬁcients through the
metering chambers and Cd;Vrepresents the overall ﬂow coefﬁcient
of the valve, with Dpvbeing the overall pressure drop through the
valve. According to Eqs. (1) and (2), it is evident that, for a given
opening degree, the ﬂow rate depends on the pressure drop
through the valve. Figure 2shows qualitatively how the metering
curve changes with the pressure drop through a valve. It refers to
an overlapped valve (namely, a valve having the spool land longer
than the adjacent gap in the valve body), which is the most com-
mon proportional valve typology , and the curve is determined
(x), given by the notch shape.
For a given opening degree, the change in the ﬂow rate because
of changes in the pressure drop can be calculated through Eq. (3),
with subscripts 1 and 2 denoting two different operating
Fig. 1 Proportional valve ATOS-DKZOR-T : 1—valve body, 2—spool, 3—solenoid, 4—LVDT, 5—electronic
control, 6 and 7—connectors)
020801-2 / Vol. 141, FEBRUARY 2019 Transactions of the ASME
It is possible to add pressure compensation by using a combina-
tion of restrictors or additional valves, so that ﬂow ﬂuctuations
due to system pressure changes are reduced. When inlet or work-
ing pressures change, the pressure compensator system keeps the
ﬂow rate constant by maintaining a constant pressure drop across
the spool oriﬁce .
Proportional valves can work in an open-loop conﬁguration or
in a closed-loop one, the latter employing a position sensor, typi-
cally a linear variable differential transformer (LVDT), for more
precise control of the spool position . Open loop control sys-
tems are cheaper, but are affected by changes in the operating
conditions, as they rely on ﬁxed parameters tuned for certain
In both cases, standard commercial electronic cards (shown in
Fig. 1) provide the solenoids with a pulse width modulation
(PWM) signal having a primary PWM signal frequency usually in
the range 200–20,000 Hz. A dither signal (square or sinusoidal
wave with a frequency lower than the PWM frequency) is also
used to keep the spool vibrating, thus overcoming stiction
between the spool and the valve body bore .
The block diagram of a typical control system is shown in
Fig. 3, adapted from Ref. . As discussed in Ref. , a
proportional–integral controller can be used for coil current con-
trol, which improves the static and dynamic characteristics of the
valve. Similar to dither, the ﬂutter signal generator has the pur-
pose of reducing both friction and the magnetic hysteresis loop of
the solenoid, improving the performance of the valve in demand-
ing applications .
The current iﬂowing through the solenoids is therefore changed
by varying the PWM duty cycle of voltage Vapplied to the sole-
noid coil, taking advantage of the resistive-inductive behavior of
the coil 
where Rand Lare the resistance and the inductance of the coil,
respectively. The higher the duty cycle of the PWM, the larger the
average intensity of the current ﬂowing through the solenoid, and
hence, the higher the electromagnetic force exerted by the sole-
noid on the spool .
The electromagnetic force (F
) acts in opposition to the damp-
ing force mainly due to the friction between the spool surface and
the valve body surface (F
x), the transient ﬂow forces and
stationary ﬂow forces (F
) due to the ﬂuid motion, and the elas-
tic force (F
¼kelx) produced by the centering springs (which are
needed to maintain the spool in a centered position when no signal
is applied to the coils). The resultant force accelerates the spool
Fact Fflow kelxc_x¼m€x(5)
Figure 4provides a representation of a section view of a typical
4/3 valve with the spool being maintained in a ﬁxed spool position
x. The overall stationary ﬂow force acting on the spool surface
along the xaxis is the sum of three contributions, due to the inter-
action between the ﬂuid and the spool within the central chamber
P-B (Fflow;center), the left chamber A-T
(Fflow;left ), and the right
chamber in correspondence of the exit T
(Fflow;rightÞ. Each com-
ponent is the sum of the pressure forces and viscous forces acting
on the spool surfaces. As analyzed in Refs. [11–13] and [23–25],
the application of the conservation of momentum to the three con-
trol volumes shown in Fig. 4leads to
Fflow ¼Fflow;left þFflow;center þFflow;right ﬃ_
mdenotes the overall mass ﬂow rate of the oil entering the
valve; ðVAÞxand ðVTÞxare the average axial velocities at the inlet
and outlet sections of the left control volume, respectively; ðVBÞx
and ðVPÞxare the average axial velocities at the outlet and inlet of
the central control volume, respectively. As the direction of the
Fig. 3 Typical control system of a proportional valve (Adapted
from Ref. )
Fig. 4 Section view of a 4/3 proportional valve (a) and enlarge-
ment on the spool surface with velocity and force vectors (b)
Fig. 2 Metering curve as a function of the pressure drop for a
given opening degree
Journal of Dynamic Systems, Measurement, and Control FEBRUARY 2019, Vol. 141 / 020801-3
ﬂow within the right control volume is orthogonal to the xaxis,
Fflow;right can be neglected. In some units, a central conical surface
(to be referred to as the compensation proﬁle) and two lateral con-
ical ones are constructed on the spool surface in order to increase
the axial velocities ðVTÞxand ðVPÞx, thus reducing the overall ﬂow
force acting on the spool .
In addition to the stationary ﬂow forces, transient ﬂow forces
are developed during the spool movement from an initial position
to a ﬁnal one. As demonstrated in Ref. , the transient ﬂow
force in a metering chamber can be calculated as
where Lis the axial distance between the inlet and outlet ports of
the metering chamber.
Like all spool valves, proportional valves are vulnerable to a
particular problem that is referred to as “hydraulic lock,” caused
by an uneven pressure distribution around the circumference of
the spool which pushes the spool radially against the inner surface
of its bore. Thus, grooves are machined circumferentially around
the spool to avoid an uneven pressure distribution and prevent
hydraulic lock .
Commercially Available Proportional Valves
Many manufacturers produce directly driven proportional direc-
tional valves, such as Atos,
Each model is usually provided by their manufacturer as a
unique body which can be equipped with different sliding spools,
according to the operation features required .
Commercially available proportional valves have less precise
manufacturing tolerances than servovalves . The larger toler-
ances on the spool geometry and spool overlap result in response
nonlinearities, especially in the vicinity of neutral spool position
A performance limitation is due to the direct actuation via pro-
portional solenoids: these are relatively heavy and can generally
operate in only one direction. Some solenoids are designed to
operate in push–pull mode, but these are more expensive and gen-
erate lower driving forces than conventional ones . In addition,
for high pressures and/or ﬂow rates required, the actuation force
generated by commercially available solenoids is not high enough
to counteract the opposing forces (ﬂow forces þelastic forces of
the centering springs). This results in a limited operational range
for these valves as far as the maximum achievable ﬂow rate is
concerned . The maximum ﬂow rate is typically 100 l/min,
with some models being capable of over 150 l/min, but only for
low pressure drops. As an example, a large valve produced by
is the DKZOR-AES model. The solenoids employed in
the DKZOR-AES model are very large and can have a maximum
input power of 50 W. The operational ﬁeld of the valve is repro-
duced in Fig. 5: it is possible to observe that the maximum ﬂow
rate is about 160 l/min, but the pressure drop must be limited to
70 bar for such a ﬂow rate level in order not to exceed the maxi-
mum power of the solenoids employed (higher pressure drop for
the same ﬂow would require a smaller metering ﬂow area and so a
higher ﬂow velocity and thus a higher ﬂow force). The increase in
the pressure drop causes a decrease in the maximum ﬂow rate
achievable; as shown by the external curve of Fig. 5, at 210 bar,
the maximum ﬂow rate through the valve is lowered to about 90
l/min. This means that, for high pressure drops, it is not possible
to reach the maximum opening degree of the valve, but only a
part of the spool stroke can be used because of the limited power
capability of the solenoids. A similar model, produced by ATOS,
namely the DHZO-AES model,
employs smaller solenoids with a
lower maximum power produced, namely up to 30 W. The internal
curve in Fig. 5reports the operational ﬁeld of the DHZO-AES; in
spite of the similar geometric characteristics, the lower actuation
power reduces the operation ﬁeld of the valve, with a maximum
ﬂow rate of 64 l/min at 210 bar.
With regard to the dynamic characteristics, they are only
slightly affected by the operating pressure, unlike two stage
valves. Available direct operated proportional directional valves
have 90 deg phase lag frequency ranging from 10 Hz to
The higher values are obtained for small valves and
those using closed-loop controls. As an example, Fig. 6shows a
reproduction of the Bode plot of the proportional valve 4WREE
size 10, produced by Bosch Rexroth.
This is a very large model,
capable of achieving 150 l/min at 100 bar pressure drop. The
Bode plot shows that the dynamic performance worsens when the
amplitude of the input signal is increased, with the 90 deg phase
lag frequency varying from 20 Hz to 40 Hz for input signal ampli-
tudes varying from 100% to 10% of the full stroke.
Similarly, the response time to a step demand can vary from
10 ms to 50 ms according to the characteristics of the unit
to the step amplitude. Figure 7shows the step tests for the valve
4WREE, achieved for 25%, 50%, 75%, and 100% of the full
Proportional directional direct operated valves are mainly used
in the industrial and transportation sectors. They are not commonly
used in aerospace, since in such application high response times
and large actuation forces are required. The latter are necessary to
avoid jammed spool conditions because of particle contamination
(chip). A proportional solenoid is not capable of providing large
actuation forces in order to shear a chip if it is jammed between the
metering edges. Such a high force level can only be obtained
through hydraulic ampliﬁcation systems (e.g., nozzle ﬂapper, jet
pipe, or deﬂector jet pilot stages) present in servovalves.
It is common practice to consider, for preliminary calculations, a
constant value of the discharge coefﬁcient for proportional valves,
with assumed values comprised between 0.65 and 0.7. However, as
highlighted in Ref. , the discharge coefﬁcient of a proportional
valve is highly dependent on the notch geometry and spool posi-
tion. In addition, the effects of cavitation are expected to affect the
discharge coefﬁcient, as discussed in Ref. .
Experimental and theoretical approaches have therefore been
used to investigate the effects of the shapes of the notches upon
the discharge coefﬁcient and exit jet ﬂow angle through a meter-
ing section of a proportional valve. In Ref. , three notch
shapes were experimentally analyzed: the ﬁrst one had a rectangu-
lar shape ended by a semicircle, the second one was obtained by
Fig. 5 Operational ﬁeld of two commercially available valves:
DKZOR-AES (external line) and DHZO-AES (internal line)
020801-4 / Vol. 141, FEBRUARY 2019 Transactions of the ASME
connecting three semicircles with very short rectangles, while the
third one had a triangular section (see Fig. 8). The experimental
circuit employed is reported in Fig. 9(a), where a pump, a variable
restrictor, a pressure relief valve, two ﬂowmeters, and two pres-
sure sensors were used to estimate the discharge coefﬁcient. The
experimental results of Ref.  showed that, for fully turbulent
ﬂow, the discharge coefﬁcient assumed different values according
to the notch type, number of notches employed, and opening
degree (spanning from 0.45 to 0.75). However, it was shown that,
for a ﬁxed opening degree, number, and typology of notches, the
discharge coefﬁcient ﬁrst increases rapidly with the Reynolds
number in the laminar ﬂow region, and then gradually achieves
the stable value for a fully turbulent ﬂow. This behavior is qualita-
tively reproduced in Fig. 10 and is similar to the graph reported in
Ref. . According to what was stated by Borghi et al. , the
pressure downstream of the restrictor was kept high enough to
avoid cavitation. Instead, the effects of cavitation upon the dis-
charge coefﬁcient of an oriﬁce were experimentally investigated
in Ref. , and the preliminary results obtained can be translated
to the metering sections of proportional valves. The experimental
apparatus, shown in Fig. 9(b), was mainly composed of an oriﬁce
interposed between two pressure relief valves which allowed the
pressure drop across the oriﬁce (Dp) to be varied along with
the upstream pressure (p
) and the downstream pressure (p
Figure 11 shows the graph of the ﬂow rate Qthrough the cylindri-
cal oriﬁce (having a ﬁxed diameter of 0.6 mm) as a function of
the square root of the pressure drop across the oriﬁce ( ﬃﬃﬃﬃﬃﬃ
addition, different curves are plotted for ﬁxed values of the pres-
sure upstream of the oriﬁce (p
). It is noteworthy that, in the ﬁrst
part of the graph, the ﬂow rate increases linearly with the square
root of the pressure drop, which means that the discharge coefﬁ-
cient remains constant according to Eq. (1). This conﬁrms the
results achieved in Ref. , namely, the discharge coefﬁcient has
a constant value for fully turbulent ﬂows. However, for each
curve, it is noted that, after a linear increase, the ﬂow rate satu-
rates reaching a constant value. This is justiﬁed by the occurrence
of cavitation, which tends to reduce the discharge coefﬁcient. In
addition, it is noted that the pressure drop at which the ﬂow rate
saturates increases with the upstream pressure p
, because the
higher the upstream pressure, the higher the downstream pressure
for a ﬁxed pressure drop. In other words, the intensity of cavita-
tion is increased by lowering the pressure downstream of an ori-
ﬁce. The graph also shows that the saturated points tend to a
linear boundary (when p
). The results obtained in Ref.
, and, in particular, the ﬂow rate trend shown in Fig. 11, are
conﬁrmed by the work carried out in Ref. . In Ref. , it is
also clearly shown that the discharge coefﬁcient of an oriﬁce, after
a constant phase as a function of the Reynolds number, undergoes
a sharp drop when cavitation occurs.
In addition to retrieving the discharge coefﬁcients, experimen-
tal approaches have also been used to measure the ﬂow forces. In
Fig. 6 Reproduction of the Bode plot of the proportional valve 4WREE size 10, produced by
Fig. 7 Reproduction of the step test diagram of the propor-
tional valve 4WREE size 10, produced by Bosch Rexroth
Journal of Dynamic Systems, Measurement, and Control FEBRUARY 2019, Vol. 141 / 020801-5
Refs.  and , the ﬂow forces were calculated as the differ-
ence between the solenoid force (F
) and the force of the center-
ing springs (F
). The electromagnetic force in a proportional
solenoid is a function of the armature position (coincident with
the spool position x) and current i. In both cases, the force surface
in the x,iplane was experimentally retrieved. In particular, the
armature-solenoid-LVDT assembly was removed from the valve
body and connected with a micrometer screw and a load cell in
order to measure the actuation force as a function of the armature
position and current, as shown in Fig. 12. Figure 13 qualitatively
shows the magnetic force surface as a function of the current and
armature position. The maximum values of the actuation forces
Fig. 8 The three notch typologies analyzed in Ref. 
Fig. 9 Experimental circuits employed in : (a), : (b), : (c), : (d), : (e), and : (f)
020801-6 / Vol. 141, FEBRUARY 2019 Transactions of the ASME
were measured to be around 100 N in Ref.  and around 140 N
in Ref. , evidencing that commercially available solenoids are
not capable of developing high actuation forces compared to ser-
vovalves, whose actuation forces can be as high as 700 N.
As an alternative approach, the actuation force was measured in
Refs.  and  by using a manual actuation system: the arma-
ture inside the coil, which is in contact with the sliding spool, is
moved through a knob; a load cell, interposed between the manual
actuation and the armature, allows the actuation force to be meas-
ured during the operation of the valve (see Fig. 14). A similar
apparatus was used in Ref.  to measure the ﬂow forces.
In addition to experimental approaches, a very effective method
for analysis of ﬂow through these valves is computation ﬂuid
dynamics (CFD), available commercially as software tools such
as ANSYS FLUENT . The ﬂow through a valve, supposed to be
incompressible, can be modeled either by setting the values of
pressure at the inlet and outlet or by setting the value of the veloc-
ity at the inlet . The use of CFD modeling has proved its
effectiveness for ON/OFF directional valves, with two-
dimensional (2D) approaches and simpliﬁed computational
domains being widely used to study the ﬂow within these valves
[37–44]. However, the ﬂow in a proportional valve is not axisym-
metrical due to the presence of notches and grooves on the spool;
for this reason, very detailed three-dimensional (3D) approaches
are commonly used to give more accurate results. Because of the
domain complexity, unstructured grids are used for proportional
valves. In Ref. , a partial 3D model (reproducing a circumfer-
ential sector of the entire valve) was used for different spool posi-
tions to study the ﬂow ﬁeld in a four-way three-position direct
operated proportional directional valve. It was demonstrated in
that paper that the use of small cylindrical notches on a spool with
spherical notches can provide ﬂow rate metering also at very
small valve opening, while not inﬂuencing the overall ﬂow forces
acting on the spool. In addition, it was shown that the compensa-
tion proﬁle (i.e., the central conical surface of the spool), if prop-
erly designed, can lead to a signiﬁcant ﬂow force reduction, with
a negligible ﬂow rate penalization at large openings. The ﬂow
force reduction is due to the increase in the axial component of
the ﬂuid velocity at the inlet section (V
, according to Eq. (6).
In Ref. , the high pressure chamber of a 4/3 proportional
valve for load-sensing applications was simulated for ﬁve spool
positions by using the open source code OpenFOAM . The
turbulence was modeled by means of the two zonal version of the
k–xmodel, known as the shear stress transport model. In particu-
lar, the effect of the direct and inverse ﬂow through the notches of
the metering chamber was investigated, showing that the dis-
charge coefﬁcient changes according to the ﬂuid direction
although a ﬁxed geometry is considered. In addition, transient
simulations were performed, in which the mesh motion was
resolved by using a generalized grid interface approach ,
Fig. 10 Qualitative trend of the discharge coefﬁcient versus
Reynolds number for a ﬁxed notch geometry [28,46]
Fig. 11 Flow rate through a ﬁxed oriﬁce as a function of the
square root of the pressure drop ( ﬃﬃﬃﬃ
pp) and upstream pressure
Fig. 12 Experimental apparatus to evaluate the actuation
forces: 1—coil, 2—LVDT, 3—load cell, 4—micrometer screw,
and 5—armature 
Fig. 13 Electromagnetic force as a function of the armature
position and current intensity
Journal of Dynamic Systems, Measurement, and Control FEBRUARY 2019, Vol. 141 / 020801-7
originally developed for turbomachinery applications and modi-
ﬁed to include not only the rotational motion of the moving grid
but also the linear displacement of a valve spool .
A partial three-dimensional stationary model was also used in
Ref.  to investigate the ﬂow characteristics of three different
groove proﬁles, namely the triangle shape, the U-shape, and the
spheroid shape, applied to a commercially available valve. The
three groove proﬁles are shown in Fig. 15, where Xis the spool
are the axial and radial cross section respec-
tively, and A
is the cross section which crosses both the throttling
edge and the lowest point of the groove. A
is the smallest cross
section across the throttling edge. Each groove has the length of
the throttling grooves in the axial direction equal to 3 mm. The
computational grid is shown in Fig. 16. An experimental circuit
was assembled to validate the results, with the use of a stepper
motor for a ﬁne adjustment of the spool position (see Fig. 9(c)).
The results conﬁrmed those obtained by Ref. , showing that
the groove shape has signiﬁcant effects on the discharge charac-
teristics, the jet ﬂow angle, the steady ﬂow force, and the throt-
tling stiffness of the spool valve . Figure 16 also reports the
pressure contours in the symmetrical surface of the notches at dif-
ferent spool positions. The pressure drop through the spheroid-
Fig. 14 Experimental apparatus for measuring the actuation force that is based on a screw
mechanism coupled with a force sensor
Fig. 15 Geometric characteristics of (a) the spheroid-shape groove, (b) the triangle-shape groove, and (c) the divergent U-
shape groove, analyzed in Ref. 
Fig. 16 Left: partial 3D CFD model employed in Ref. ; right: pressure contours on the symmetry plane: (a)X50.6 mm, (b)
X51.4 mm, and (c)X52.0 mm 
020801-8 / Vol. 141, FEBRUARY 2019 Transactions of the ASME
shape groove is concentrated on cross sections A
the entire range of the spool stroke, and with the increase of the
opening, the proportion of the pressure drop changes gradually
from cross sections A
. For a triangular notch, the pressure
drop is mainly centralized in cross section A4 . The divergent
U-shape groove has a more complex behavior of the pressure dis-
tribution than the other two types, with the pressure drop distribu-
tion changing remarkably according to the spool position.
The experimental and numerical activity also allowed the con-
stant values of the discharge coefﬁcient to be calculated for fully
turbulent ﬂow (see Fig. 10), and these values are reported in
Table 1. Figure 17 shows the values of the jet ﬂow angles obtained
via CFD for different shapes of the grooves analyzed. It is note-
worthy that very different values are obtained and that the opening
degree also signiﬁcantly affects the discharge coefﬁcients and
ﬂow angles . This analysis is instrumental in pointing out the
importance of CFD, which can allow a precise evaluation of the
ﬂow characteristics of a proportional valve for a given geometry
of the spool notches.
A 3D CFD model reproducing a part of the spool surface was
used to investigate the effects of other important geometrical fea-
tures , such as the circumferential grooves machined on the
spool surface. As only the zone in correspondence of the circum-
ferential grooves was simulated, quadrilateral cells were gener-
ated. Circumferential grooves are fundamental to avoid hydraulic
lock caused by an uneven pressure distribution on the spool sur-
face during its movement. Hong and Kim suggested using spiral
grooves instead of typical circumferential ones. Their work dem-
onstrated that spool valves with spiral grooves could offer better
performance in terms of relieving the asymmetric pressure distri-
bution in the radial clearance because spiral grooves act as one
continuous groove .
With the ever-increasing capability of computer hardware
resources, the use of fully 3D approaches has become more com-
mon to study these valves via CFD [32,33,47–51]. Full 3D model-
ing was used in Ref.  to conﬁrm at ﬁrst that the use of
constant values for the discharge coefﬁcient may lead to signiﬁ-
cant errors and then to obtain a method for calculating the coefﬁ-
cient values. Their numerical results were validated through the
experimental circuit shown in Fig. 9(d). In particular, functions
for evaluating the ﬂow coefﬁcient were proposed that depend on
the spool position and on the ﬂow rate. These functions can be
particularly useful at the design stage in order to properly design
the spool surface for a given metering curve.
The full 3D CFD analysis of Ref.  was carried out at the
maximum opening of the WE10H valve produced by Bosch
Rexroth to predict the stationary ﬂow force, and this was calcu-
lated as a function of the ﬂow rate. Figure 18 shows the computa-
tional grid, whereas Table 2reports the values of the predicted
ﬂow forces, extrapolated from Ref. . The numerical results
were also in very good agreement with experimental data, thus
conﬁrming the high accuracy reached by current full 3D methods.
A full 3D method was also used in Ref.  to develop a pressure
compensation method for multisection proportional directional
control valves which is based on the adjustment of the forces act-
ing on the spool and does not need the use of additional compen-
sating valves or other correcting elements, such as sensors in a
feedback control system .
Table 1 Asymptotic value of the discharge coefﬁcient according to different notch proﬁles
Discharge coefficient (asymptotic value)
Opening (mm) Spheroid shape groove Triangle shape groove Divergent U-shape groove
0.6 0.747 0.720 0.651
1.4 0.682 0.692 0.646
2.2 0.620 0.666 0.652
Adapted from Ref. .
Fig. 17 Values of the jet ﬂow angle according to different
notch proﬁles obtained via CFD 
Fig. 18 Full 3D grid developed in Ref. 
Journal of Dynamic Systems, Measurement, and Control FEBRUARY 2019, Vol. 141 / 020801-9
A further example of full 3D modeling is reported in Ref. .
In that work, the aim was to increase the accuracy in the predic-
tion of the stationary ﬂow rate and ﬂow forces compared to par-
tially 3D models; 11 spool positions covering two-thirds of the
full stroke were simulated for a commercially available valve, and
11 unstructured meshes with about two million cells were gener-
ated. The RNG k–emodel coupled with the enhanced wall treat-
ment was implemented to resolve turbulence. The numerical
predictions were compared with experimental results obtained
through an experimental hydraulic circuit. The paper showed that
a full 3D discretization of the entire ﬂow within the valve is
required to properly predict the ﬂow at small openings, where an
axisymmetric approach fails, and in particular at large spool dis-
placements, where also a partially 3D discretization had shown its
limitations in previous papers.
A full 3D model was also employed in Ref.  to evaluate the
effects of small cylindrical notches to be machined on the spool
surface of a commercially available valve manufactured by
Two spool versions with one notch and with
two symmetrical notches were considered. Computational grids,
each composed of about 4.1 million cells, were generated for gap
widths spanning from 0.1 mm to 0.4 mm with a step of 0.1 mm.
The results conﬁrm that the use of small cylindrical notches at the
apex of main grooves can allow a proportional valve to operate
with very low ﬂow at small openings. In addition, it was demon-
strated that the use of two notches arranged symmetrically on both
sides of the spool determines radial force compensation.
A further improvement in the CFD modeling of proportional valves
was given in Ref. , where the employed CFD model also
accounted for cavitation, which is a non-negligible effect occurring in
these valves. Among the available cavitation models, the Schnerr and
Sauer model provided by FLUENT was chosen since it is very robust
and converges quickly. The results provided by the cavitation model
were compared with the results obtained by the monophase model (in
which the ﬂuid was treated as incompressible). An experimental
hydraulic circuit, shown in Fig. 9(e), was also assembled in order to
evaluate the effectiveness of the numerical model. A pressure relief
valve was placed downstream of the proportional directional valve
which allowed the pressure to be increased at port T, in order to evalu-
ate the performance of the proportional valve without cavitation (due
to the high discharge pressure). In contrast, cavitation could be gener-
ated by opening a block valve so that the hydraulic oil was able to by-
pass the pressure relief valve, and the pressure at port Twas decreased
nearly to the atmospheric value . Figure 19(a)shows the full 3D
grid employed in the simulations. The “porous jump” boundary con-
dition allowed a ﬁxed pressure variation to be assigned through sec-
tion C (see Fig. 19(a)), so as to simulate the pressure drop registered
through the measuring equipment in the experimental tests. The pres-
sure drop through the porous surface is computed by FLUENT as
Dpporous ¼ l
where dis the permeability of the medium, cis the pressure-jump
coefﬁcient, vis the velocity normal to the porous face, qand lare
the density and the molecular viscosity of the ﬂuid, respectively,
and Dmis the thickness of the medium .
Figure 19(b)shows the metering curves obtained experimen-
tally and numerically for low discharge pressure and for high dis-
charge pressure maintaining an overall pressure drop of 70 bar. It
can be seen that cavitation affects the ﬂow rate through the valve,
causing a ﬂow rate reduction of about 8% at the maximum open-
ings. The ﬂow rate reduction is due to the reduction of the dis-
charge coefﬁcient in the low pressure chamber (metering section
B-T in Fig. 19(a)), where cavitation occurs. The experimental
results were very close to the numerical predictions, thus demon-
strating that such a CFD model can reliably predict cavitation.
Figure 19(c)provides the contours of the vapor volume fraction
computed on the spool surface and on a section plane in corre-
spondence of metering chamber B-T for the spool displacements
equal to x¼0.8 mm and x¼1.4 mm, highlighting the importance
of the phenomenon, especially at the large openings.
In Ref. , a full 3D CFD analysis was performed to study a
new concept of valve. The solution presented in that paper uses an
axial ﬂow valve, where the oil passes through the valve along its
axis, with two rotating surfaces causing a rotational metering. The
result of that new design approach shows several advantages with
respect to the common spool valves, such as the extremely com-
pact size and the device versatility. This particular valve can real-
ize the majority of the functions achievable using a two-way
two-position proportional valve piloted by two pressure signals
(for example, a pressure compensated valve); the axial ﬂow and
the “built-in” metering edges yield the possibility to produce this
valve as a cartridge component . Other examples of new con-
cepts of proportional valves with metering features obtained
through the rotation of the spool and investigated via full 3D CFD
models are present in the literature, such as Refs. [54–56].
Such detailed 3D models can also be used at the design stage to
obtain very effective geometries for the spool and for the valve
body of standard valves in order to reduce the ﬂow forces or cavi-
tation intensity. In this regard, the current research studies regard-
ing spool valves are focused on reducing the ﬂow force to extend
their application range [11,23–25,34,57]. As pointed out in Ref.
, commercially available valves present nonoptimized geome-
tries which restrict their potential, and opportune geometrical
modiﬁcations to the valve body and spool are needed to minimize
the stationary ﬂow forces, which play a more important role in the
control of higher hydraulic powers compared to the other resistant
forces. In Ref. , effective changes were made both to the slid-
ing spool and to the valve body of an ON/OFF small hydraulic
seat valve, conﬁrming that the nonoptimized proﬁles of commer-
cially available valves have a great inﬂuence on the required
actuation forces. In Ref. , some possible methods were also
provided to reduce the static ﬂow forces in sliding-spool valves:
the results of this research are very promising and prove that the
axial component of the ﬂow forces, and therefore, the necessary
actuation force can be reduced signiﬁcantly just by modifying the
geometry of the valve housing and spool.
In Ref. , a genetic algorithm was coupled with a full 3D
model of the ﬂow ﬁeld of a proportional valve in order to reduce
the ﬂow force at the maximum opening for a commercially avail-
able valve. The geometrical parameters of the valve body were
kept unchanged, whereas four geometrical parameters of the valve
spool were selected as design parameters. The parameters, shown
in Fig. 20, deﬁne the central and lateral surfaces of the spool,
allowing the velocities at the inlet and outlet selections of the
valve to be varied according to Eq. (6).
The comparison between the reference values and optimized
ones is shown in Fig. 20. The optimized spool was constructed
and experimentally compared with the reference one in Ref. .
A manual actuation system coupled with a load cell was used to
measure the actuation force required by the two spools (see
Figs. 9(f)and 14). An actuation force reduction of about 13% at
the maximum opening was measured for a pressure drop of 70 bar
through the valve.
Table 2 Flow force predicted at the maximum opening for a
commercially available valve as a function of the ﬂow rate
Flow rate (l/min) Flow force (N)
Adapted from Ref. .
020801-10 / Vol. 141, FEBRUARY 2019 Transactions of the ASME
Fig. 19 Full 3D grid used in Ref. (a), metering curves obtained numerically and experimentally for low and high discharge
pressure with an overall pressure drop 570 bar (b), and contours of volume fraction on the spool surface (c)
Journal of Dynamic Systems, Measurement, and Control FEBRUARY 2019, Vol. 141 / 020801-11
As an alternative approach for the ﬂow force reduction, Lisow-
ski et al.  suggested introducing additional channels in the
valve body, without changing the spool geometry, as shown in
Fig. 21. In that paper, it was noted that the reduction of average
velocity around the spool and better pressure compensation in the
valve body are associated with lower ﬂow force acting on the spool
. The numerical and experimental comparison between the ref-
erence valve and the novel one showed that in the latter the pressure
is more balanced around the spool. A very large ﬂow force reduc-
tion, up to 50%, was obtained with the additional channels.
The main settings of some of the 3D simulation studies ana-
lyzed so far that predict the ﬂow ﬁeld through a proportional
directional valve are summarized in Table 3.
Research on Control Systems
The literature review analyzed so far has been concerned with
the ﬂuid dynamic behavior of proportional valves. In parallel to
this, many studies have considered improving the spool position
control systems for these valves, often with the help of models of
the dynamic characteristics of spool actuation. These models often
take advantage of software packages capable of studying the valve
dynamics, such as MATLAB , SIMULINK , and AMESIM [60,61].
A nonlinear dynamic model was developed in Ref. , in
which the solenoid was modeled as a nonlinear resistor/inductor
combination, with inductance parameters changing according to
the values of displacement and current. Empirical curve ﬁtting
techniques were used to model the magnetic characteristics of the
solenoid, enabling both current and magnetic ﬂux to be simulated.
The spool assembly was modeled as a spring/mass/damper sys-
tem. The inertia and damping effects of the armature were incor-
porated in the spool model. The solenoid model was used to
estimate the spool force in order to obtain a suitable damping
coefﬁcient value. The model accurately predicts both the
dynamic and the steady-state response of the valve to voltage
Fig. 20 Design parameters adopted for the ﬂuid dynamic optimization performed in Ref.  and experimentally validated in
Ref. : comparison between reference geometry and optimized one
Fig. 21 Contours of pressure in the novel valve body geometry presented in Ref. : 2 and 3 denote the
020801-12 / Vol. 141, FEBRUARY 2019 Transactions of the ASME
Analysis of a proportional solenoid was performed using ﬁnite
element (FE) simulation in Ref. , by adopting an axially sym-
metrical two-dimensional (2D) FE model in ANSYS/EMAG. The
electromagnetic force and ﬂux linkage characteristics in all arma-
ture positions and for different currents were analyzed. Also in
Ref. , a FE model was used to develop a nonlinear model of
various types of proportional valves, but in this case a full 3D
model was developed and experimentally validated by the
authors. The 3D model allows quantifying the effects of the eddy
currents and retrieving a second-order transfer function which
describes the electromagnet dynamics. The developed nonlinear
model was composed of three submodels based on a lumped
parameter approach: the ﬂuid-dynamic model (for the evaluation of
the main ﬂow features), the mechanical model (which solves the
mobile body motion), and the electromagnetic model (which evalu-
ates the magnetic forces and the electric transient). The comparison
between the nonlinear model and the linear model shows the limits
of the linear approximation to study the real components .
In Ref. , a discontinuous projection based adaptive robust
controller was developed that is capable of compensating for the
effect of the valve deadband, and certain straight-line approxima-
tions were used to model the nonlinear ﬂow gain coefﬁcient of the
valve. With regard to the analysis of the deadband in these valves,
which is a key factor that limits both static and dynamic perform-
ance in feedback control of ﬂuid power systems, in Refs.  and
, a new methodology for the identiﬁcation of the dead zone
was proposed. The proposed method was based on the observation
of the dynamic behavior of the pressure in the valve gaps and was
achieved by using only pressure transducers. Experimental tests
were carried out to demonstrate the efﬁcacy of this methodology.
A novel nonlinear sliding mode controller was developed in
Ref. . The results demonstrated that the sliding mode control-
ler can determine fast response times, with small overshoots and
In Ref. , a multidomain nonlinear dynamic model of a pro-
portional solenoid valve system was developed in the form of non-
linear state equations and was validated by experimental data.
This model successfully predicts the dynamic characteristics of
the valve and can be used as a powerful computational and simu-
lation tool for valve design and algorithm optimization .
In Ref. , a control strategy that is based on the peak and
hold (P&H) technique and that requires only a low cost microcon-
troller was proposed. The P&H technique, widely used to control
Diesel fuel injection systems, consists of a particular PWM signal
with a variable duty cycle composed of two constant signal phases
with different voltage and current values. Similarly, the system
proposed in that paper for the control of proportional valves
employs, after the polarization phase, a peak high duty cycle to
approach the target position of the spool followed by a lower duty
cycle (hold) to maintain the position (see Fig. 22). This strategy
gives a very fast valve response, even comparable to that provided
by standard closed-loop control systems, with a cost similar to
available open-loop control systems .
In Ref. , a digital state observer feedback control system,
which is based on the digital signal processor and dynamic mathe-
matical model of a proportional valve, was designed. Bu and Yao
 proposed three different types of controllers to improve the per-
formance of proportional directional valves: (a) an open-loop com-
pensator which requires the accurate valve dynamic model
information; (b) a full state feedback adaptive robust controller, which
effectively takes into account the effect of parametric uncertainties
and uncertain nonlinearities such as friction force and ﬂow force; (c)
an output feedback adaptive robust controller to address the problem
of unmeasurable states which takes into account the effect of both
parametric uncertainties and uncertain nonlinearities .
A new method to tune the proportional-integral-derivative
(PID) parameters of controllers for proportional directional valves
without modeling and a priori knowledge of the system was
Table 3 Some partially and fully 3D models presented in the scientiﬁc literature that simulate the ﬂow ﬁeld through a proportional
Authors CFD software Simulations Domain Number of cells Fluid model
model Wall treatment
Amirante et al.
ANSYS FLUENT Full stroke,
drop ¼40 bar
Milani et al.
OpenFOAM Five spool
SST k–xNot mentioned
Lisowski et al.
ANSYS FLUENT Fixed position,
30, 60, 90, 120
and 150 l/min
Amirante et al.
ANSYS FLUENT 2/3 of the full
drop ¼100 bar
Ye et al. ANSYS FLUENT Five positions
of the full
Amirante et al.
ANSYS FLUENT Full stroke,
drop ¼70 bar
according to the spool
Lisowski et al.
ANSYS FLUENT Gap widths
0.1 mm to
0.4 mm with a
step of 0.1 mm
4,100,000 cells Incompressi-
Journal of Dynamic Systems, Measurement, and Control FEBRUARY 2019, Vol. 141 / 020801-13
proposed in Ref. . The optimization of the PID parameters was
achieved through statistical analysis by using the proposed method.
The results showed that the optimized controller performs well, con-
ﬁrming that the tuning method is effective while also being straight-
Proposals to employ controllers based on Fuzzy logic are also
present in the scientiﬁc literature. In Ref. , an unconventional
electro-hydraulic proportional ﬂow control valve based on a
switching solenoid and a fuzzy-logic controller was proposed for
application to hydraulic presses. The fuzzy-logic controller was
employed to linearize the force/stroke characteristics. The experi-
mental results showed very good results in terms of the control of
the ram velocity of a press cylinder. A combined fuzzy-PID sys-
tem was designed in Ref.  to control a proportional valve,
showing fast response and small overshoot.
In Ref. , a control method was proposed that uses the pro-
portional solenoids simultaneously, contrary to the normal control
method that energizes only one solenoid at a time. The perform-
ance of the valve is greatly inﬂuenced by the nonlinearity of the
proportional solenoid, such as dead zone and low force gain with
a small current, and this effect cannot be eliminated by a simple
dead-zone current compensation . To avoid this disadvantage,
the authors of that paper proposed a differential control method
(DCM). By employing DCM, the controller outputs differential
signals to simultaneously energize both solenoids of the propor-
tional valve, and the operating point is found by analyzing the
force output of the two solenoids to minimize the variation of the
current-force gain. The comparisons of the valve response charac-
teristics were performed between normal control method and
DCM by nonlinear dynamic simulation and experiments. Simula-
tion and experimental results showed that, by using DCM, the fre-
quency response of the valve is greatly enhanced, especially when
the input is small, which means that the dynamic characteristics of
the proportional valve are improved .
In Ref. , severe nonlinearities around the spool neutral posi-
tion due to nonlinear spring force, nonlinear solenoid force and
disturbances arising from ﬂow force working on the spool were
overcome through a dead-zone compensation design, a disturb-
ance rejecting control design, and a control design for improving
the reference tracking ability. In particular, an input shaping ﬁlter
was applied to optimize the control characteristics in the high fre-
quency range . The results demonstrated that the application
of the proposed control design using input shaping ﬁlter can satis-
factorily compensate for the dead-zone and greatly improve the
dynamic response characteristics of the valve .
In Ref. , relationships of classical electrodynamics were
used to derive a clear and detailed model that describes the inﬂuence
of sinusoidal currents on the inductance and eddy current resistance,
dependent on the spool position. In addition, a control theoretic
approach was proposed to determine the spool position, and a
numerically efﬁcient reduced order observer was designed .
In Ref. , the application of a model-based control structure
called embedded model control was presented. The overall control
consists of two hierarchical loops: the inner loop is the solenoid
current regulator with a closed-loop bandwidth close to 1 kHz.
The outer loop is a position tracking control, in charge of the
accurate positioning of the spool with respect to valve openings
In Ref. , a method for identifying solenoid valve transition
events by analyzing the current through the solenoid coils was
proposed. The method estimates the spool position through identi-
fying slope changes in the solenoid coil current traces. This meth-
odology was able to identify the timing of valve transition events
with less than 7% error compared to the measurement of the posi-
tion of the valve spool obtained through a laser displacement sen-
sor. As the method is based on measuring the current, it requires
no modiﬁcation to the valve or valve housing .
In Ref. , an improved nonlinear sliding-mode controller
was developed. Experimental studies were conducted and the
results showed that the controller can achieve a continuous and
stable sliding mode state, realize the time-optimal step response
of a valve, and exhibit strong disturbance rejection abilities .
In Ref. , the detection of faults in these valves was
addressed. It was shown that faults could be detected with the use
of additional hardware or software. A proposal for low-cost, efﬁ-
cient changes to commercially available valves was also given in
A new idea is represented by the “digital hydraulics” approach,
which aims at applying digital principles to hydraulics by using
multiple or high-speed on/off valves controlled through software
rather than using proportional and servovalves [80,81]. With
regard to multiple valves, the idea is to use only one size of valve,
which is optimal in the sense of ﬂow density, response time, and
fault tolerance. However, this concept is very demanding, as a
larger number of valves are needed. The beneﬁts of such systems
are fast and amplitude independent response, redundancy, and
robustness against oil impurities . Drawbacks are large space
requirement, complexity of control, and possibly higher price.
Several papers are present in the scientiﬁc literature that study
digital hydraulics systems in detail [49,82–89].
This paper has provided an overview on the operating princi-
ples, mathematical modeling, industrial and research state of the
art of directly driven proportional directional hydraulic valves.
These valves contain an inner sliding spool, directly moved by
solenoids inside a valve body, which is provided with notches to
allow ﬂow rate metering as a function of the spool position. Com-
mercially available units present lower dynamic performance but
also lower costs than servovalves. The operational ﬁeld on the
ﬂow rate-pressure plane is limited because sufﬁciently compact
solenoids are not capable of counteracting high ﬂow forces.
The ﬂow rate and ﬂow forces can be predicted by means of sim-
ple formulae which can be very useful for preliminary calcula-
tions. The validity of these formulae has been widely
demonstrated; however, they depend on the discharge coefﬁcient
and ﬂow angles, which can assume different values according to
the spool position, notch geometry, and operating conditions. To
investigate how these parameters affect the discharge coefﬁcient
and ﬂow angle, experimental and CFD approaches have been used
in the scientiﬁc literature. The role of CFD modeling is notewor-
thy as it can allow a precise evaluation of the ﬂow characteristics
of a proportional valve for a given geometry of the spool notches,
without the necessity of setting up an entire experimental circuit.
At ﬁrst, authors concentrated their efforts in developing partially
3D CFD models. However, with the ever-increasing capability of
hardware resources, the use of fully 3D approaches has become
more common, and the scientiﬁc literature presents several papers
in which the high accuracy of full 3D models is demonstrated by
comparing numerical predictions with experimental data. The
ﬂow is usually treated as incompressible, however some models
also take into account the occurrence of cavitation by means of a
Fig. 22 Peak and hold technique employed in Ref. 
020801-14 / Vol. 141, FEBRUARY 2019 Transactions of the ASME
mixture model which simulates the formation of vapor inside the
liquid. The use of CFD to predict the discharge coefﬁcient, ﬂow
angle, and occurrence of cavitation can assist valve design by
studying optimized geometry, in particular to allow the ﬂow
forces to be reduced, thus aiming at enlarging the operation ﬁeld
of these valves. A few optimized geometries are proposed in the lit-
erature that are more effective than commercial ones in terms of
ﬂow forces. These geometries regard either the optimization of the
spool proﬁle, by acting on the inlet and outlet velocity angles, or
the valve body, by adopting additional channels which can equalize
the circumferential pressure distribution on the spool.
Research has also been focused on improving the control sys-
tems of these valves. Detailed models of the solenoid assembly
are proposed to accomplish this task. Several effective methods
have been proposed, such as a control based on the peak and hold
technique, or a differential control method in which the controller
outputs signals to simultaneously energize both solenoids of a pro-
portional valve to improve linearity, or the employment of shap-
ing ﬁlters or hierarchical control loops, or sliding mode
controllers. A new idea is digital hydraulics, which aims at apply-
ing digital principles to hydraulics by using multiple or high-
speed on/off valves controlled through software rather than using
proportional or servovalves.
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