Nonlinear Modelling, Dynamic Performances and control of an
Electrical Syringe Pump Using a Linear Actuator
Abstract: - This article discusses the non-linear improved modelling approach, dynamic performances with and
without saturation and closed loop control of an electrical syringe pump using a linear actuator. This work is
intended for the mathematical characterization and setting equation of LSRM including the effect of magnetic
nonlinearities. The work is beginning by the modelling of the inductance variation versus current. This variation
is analytically described using Fourier series and the polynomial functions are used to express the nonlinear
variation of its coefficients versus current. The second step is reserved to the prediction of the dynamic
performances, taking into account the saturation phenomenon. Finally, the linear stepper motors movement is
characterized by a highly oscillatory translation, which is troublesome for the positional accuracy and the speed
constant such as depend by many industrial applications such as the syringe pump. Thus, in order to attenuate
the amplitudes of these oscillations and guarantee the positioning of the actuator without errors, solutions
exploiting closed-loop control techniques are proposed in this paper for the purpose of improve the actuator
performances.
Key-Words: - linear motor, electrical syringe pump, nonlinear inductance, Fourier series, FEM, saturation.
Received: October 14, 2022. Revised: April 24, 2023. Accepted: June 5, 2023. Published: July 7, 2023.
1 Introduction
The syringe pump is a conventional biomedical
system hospital emergency service. It is mainly used in
intravenous, intra-arterial infusions, anesthetic infusions
and chemotherapy. It is essential in the various general
surgery departments and the internal department of
medicine for children, adults, pediatrics, the emergency
department, gynecology, and soon.
Perfusion is a continuous drug delivery method
over time at a constant rate. Whatever the nature of the
disease, the infusion treatment is all the more effective
that the injected doses are balanced and regularly
distributed over time.
The satisfaction of this requirement needs the use
of syringe pumps whose technology is constantly
evolving in search for performance improvement, [1-2].
Motorization of this electrical syringe pump is often
ensured by a Linear Stepping Motor.
Nowadays, LSRM are widely used. Unfortunately,
in order to generate a high-propulsion force the LSRM
must be operated in the saturation zone. In saturation
conditions, main magnetic characteristics, such as flux
linkage, inductance and propulsion force, are highly
nonlinear. Modeling and simulation of a switched
reluctance machines is more complex than that of AC
and DC motors due to its highly non-linear operation.
These nonlinearities are introduced by two main
factors: Magnetic saturation and the air gap variation.
Consequently, the model based on some hypotheses are
not very accurate to determine system performances
and elaborate control strategies, [3- 4].
The LSRM is always operating in the magnetically
saturated mode to maximize the energy transfer. In a
LSRM, the phase inductances and flux linkages vary
with rotor position due to stator and rotor saliencies.
The phase inductances and flux linkages at any rotor
position also vary with the instantaneous phase currents
because of magnetic saturation. However, these
variations can be modelled analytically using the
obtained data through FEM or via experiments. These
analytical expressions are used to represent the LSRM
dynamics and hence the machine performances can be
obtained, [5-6-7-8-9-10].
There has been widespread research in the Linear
Switched Reluctance Motor modelling area, which
includes the effect of magnetic nonlinearities. However,
there is still a lack of accuracy. For this purpose, this
paper provides an improvement of the classical
approach formerly known, [11-12-13-14-21].
The paper is organized as follows. Taking apart the
introduction and the conclusion, section 1 gives the
dimensional characteristics of the used motor and
IMED MAHMOUD1,3, 2,3ADEL KHEDHER
1Department of Electrical Engineering, Higher Institute of Applied Science and Technology Mahdia, TUNISIA
2Department of Electrical Engineering, National Engineering School of Sousse, TUNISIA
3Laboratory of Advanced Technology and Intelligent Systems (LATIS), TUNISIA
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.7
Imed Mahmoud, Adel Khedher
E-ISSN: 2769-2507
48
Volume 5, 2023
syringe. Section 2 details the proposed improved model
and compares to the results extracted from the FEM.
Section 3 is reserved to determine the dynamic
performances including the magnetic nonlinearities
effect. Finally, Section 4 is to suggest closed loop
control approaches taking into account the dynamic
behavior ensuring precise positioning and without
overshoot for different medication infusion rates.
2. Dimensional characteristics of the all
biomedical system.
The stepping linear motor, thanks to its reliability
and control simplicity, is used in various applications
such as robotics, machine tools ordering, space
applications and military and data-processing
peripherals. In the medical field, it is largely used as
well for the displacement and the shunting of the
surgical laser beams as for medical scanners piloting
apparatuses. The Descriptive diagram of the all
biomedical system, to be studied, is illustrated by
Figure 1 where all geometrical parameters are defined.
Some mechanical and electrical parameters of the
actuator and syringe are given in table 1 (appendix).
Fig.1. Block diagram representation of the biomedical system
De
Syringe
x
Lp
Lt
Ls
Lc
Dc
Dt
Ds
Fm
Cylinder
Slide
Tube
Lr
Hs
Ls
a
b
c
Hr
D
mover
Stator
Figure 1. Block diagram representation of the biomedical system
3. Analytical model of the LSRM
Figure 2. Classification of LSRM modelling
In order to determine the non-linear improvement
modelling approach which describes the behavior of a
saturated reluctant structure, there are basically two
ways to represent the static LSRM characteristics. The
first is to plot the phase flux linkage variations with
rotor position and phase current. The second one is to
plot the phase inductance variation with rotor position
at different phase currents. These static characteristics
are highly nonlinear. Figure 2 shows a classification of
the different LSRM modelling techniques.
In LSRM, the magnetic reluctance is strongly
affected by the nonlinear characteristic of the employed
magnetic matters. Consequently, this variation is
reflected on the inductance evolution. Accordingly, the
phase inductance changes periodically as function of
the rotor position. At any given rotor position, the
phase inductance also varies with the instantaneous
phase current as shown in the figure 3. It is maximum
in the aligned position and minimum in the unaligned
position.
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.7
Imed Mahmoud, Adel Khedher
E-ISSN: 2769-2507
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Volume 5, 2023
-5
0
5
0
0.5
1
0.05
0.1
0.15
Displacement (mm)
Current (A)
Inductance (H)
Figure 3. Response surface of the inductance as
a function of position and current.
Therefore, the phase inductance versus mover position
will be represented by Fourier series (1) and the
nonlinear variation of its coefficients with current will
be expressed by polynomial functions (6, 7 and 8), [16-
18-23-24-26]:
0
( , ) ( )cos
m
kr
k
L x i L i kN x
(1)
With
i xandm,
are respectively the phase current, the
mover position and the number of Fourier series terms.
The numerical simulations accuracy and stability
are the main challenges which should be met. To
simplify Eq (1), by considering the first three terms of
the Fourier series are considered. The inductance
expression is given by Eq (2), [15-17-20-21]:
j j j r
r
jr
r
L x i L i L i N x j NN
L i N x j NN
01
2
2
( , ) ( ) ( )cos( ( ( 1) ))
2
( )cos(2 ( ( 1) ))
2)
With
j
L x i( , )
is the inductance associate to the phase
j
in the mover position
x
for the current
j
i
and
N
the
phase number.
The analytical model can be developed by
employing a curves number depending on the Fourier
series order. A second order Fourier series requires
three positions to determine three coefficients L0, L1
and L2. The useful operation of the actuator extends
from the starting position x = 1.5 mm to the position of
conjunction x = 0 mm. It follows that the description of
the indispensable intermediate position, such that x =
0.75mm.
To determine the three coefficients
L L and L
0 1 2
,
, we
use the inductance at three positions:
L x mm( 0 )
,
L x mm( 0.75 )
and
L x mm( 1.5 )
.
Therefore, these three coefficients are written as
follows:
0
12
( (0)+ (1.5))- (0.75)
2
21
L L L L
(3)
1
1(0)- (1.5)+2 (0.75)
21
L L L L
(4)
2
1 2 2
(0)+ 1- (1.5)- (0.75)
22
21
L L L L
(5)
The inductance
Lix mm
() 0
,
Lix mm
() 0.75
and
Lix mm
() 1.5
are function of the phase current and can
be approximated by polynomials as follows:
0
( 0)
p
n
i j n j
n
L x a i
(6)
0
( 0.75)
p
n
i j n j
n
L x b i
(7)
0
( 1.5)
p
n
i j n j
n
L x c i
(8)
With p is the order of the polynomials and an, bn, cn
their coefficients. In our study, p=6 was considered
sufficient for an adequate description, Figure 4.
Figure 4. Inductance against phase current for three
positions.
The inductance curve versus current in position
0x mm
was approximated using the following
polynomial:
6 5 4 3 2
6 5 4 3 2 1 0
0
( )x mm a i a i a i a i a i aL ii a
(9)
In a similar way, we find:
6 5 4 3 2
6 5 4 3 2 1 0
0.75
( )x mm b i b i b i b i b i b i bLi
(10)
6 5 4 3 2
6 5 4 3 2 1 0
1.5
( )x mm c i c i c i c i c i c i cLi
(11)
0 0.2 0.4 0.6 0.8 1 1.2
0.06
0.08
0.1
0.12
0.14
0.16
Courant (A)
Inductance (H)
+ FEM
+ FEM
0x
0.75x mm
1.5x mm
Current (A)
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.7
Imed Mahmoud, Adel Khedher
E-ISSN: 2769-2507
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Volume 5, 2023
The polynomial description of functions
Lix mm
() 0
,
Lix mm
() 0.75
and
Lix mm
() 1.5
in Eqs (9), (10) and
(11) gives the functions
L i L i and L i
0 1 2
( ), ( ) ( )
described
in Eqs (3), (4) and (5) so that:
i d i d iLi idd d i d d i
65432
6 5 4 3 2 1 00 ()
(12)
i e i e iLi iee e i e e i
65432
6 5 4 3 2 1 01 ()
(13)
i h i h iLi ihh h i h h i
65432
6 5 4 3 2 1 02 ()
(14)
All coefficients of Eqs (9), (10), (11), (12), (13)
and (14) are given in Tables 2 (Appendix).
Figure 5 shows the comparison between the
inductance characteristics obtained by the improvement
proposed model and the obtained result through FEM.
Figure 5. Comparison of the inductance characteristics
against phase current for three positions.
In addition, it is to be noted that by multiplying the
expression of the inductance by the current one derives
that of the flux.
xx
i L i i
(15)
The stator phase inductance at the aligned position is
very affected by the stator phase current variations. On
the contrary, the unaligned inductance is practically
constant due to the large reluctance that characterizes
this position.
It is worth mentioning that, the found analytical
model remains valid for any position “x” and any
current ”i” as illustrated by figures 5 and 6. These
figures provide better characteristics predetermination.
Figure 6. Flux characteristics Versus current for
different positions.
4. Dynamic behaviour of the biomedical
application
The actuator is coupled with the syringe plunger in
presence of the aqueous solution to perfuse. We plan to
study the dynamic behavior of all biomedical
application. It is well known that the total
electromagnetic force is given by the following
expression, [19-22-25-26]:
,0
,
,
i
j j j
cj
j
L x i i di
W
F i x xx
(16)
For a given current, equation (16) becomes:
2
()
1
( , ) 2
j
jj
i cte
Lx
F i x i
x
(17)
After development we get:
12
00
2 2 4 4
, sin( ) ( ) sin( ) ( )
ii
F i x x L i i di x L i i di
(18)
With:
8
12
2
0
1
()
i
k
k
k
L i i di e i
k
(19)
8
22
2
0
1
()
i
k
k
k
L i i di h i
k
(20)
Remember that the coefficients
k
e
and
k
h
are
defined previously. Finally the total electromagnetic
force is given by:
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Current (A)
Flux linkage (Wb)
x=0 mm
x=0.75 mm
x=1.2 mm
x=1.5 mm
x=1.8 mm
x=2.1 mm
x=2.4 mm
x=3 mm
Model
* FEM
0x
3x mm
0 0.2 0.4 0.6 0.8 1 1.2
0.06
0.08
0.1
0.12
0.14
0.16
Courant (A)
Inductance (H)
Current (A)
0x
0.75x mm
1.5x mm
Model
* FEM
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DOI: 10.37394/232027.2023.5.7
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E-ISSN: 2769-2507
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8
2
2
8
2
2
1 2 2
, sin( )
1 4 4
sin( )
k
k
k
k
k
k
F i x e i x
k
h i x
k
21)
The four phases of LSRM are described by these
dynamic electric equations as follows:
( , ) ( , )
( , ) A A A A
A A A A
A
di L x i L x i di
dx
U Ri L x i i
dt x dt i dt
(22)
( , ) ( , )
( , ) B B B B
B B B B
B
di L x i L x i di
dx
U Ri L x i i
dt x dt i dt
(23)
( , ) ( , )
( , ) C C C C
C C C C
C
di L x i L x i di
dx
U Ri L x i i
dt x dt i dt
(24)
( , ) ( , )
( , ) D D D D
D D D D
D
di L x i L x i di
dx
U Ri L x i i
dt x dt i dt
(25)
For the given phases the derivative of the inductances is
given by:
12
( , ) 2 2 4 4
( )sin( ) ( )sin( )
L x i L i x L i x
x
(26)
' ' '
0 1 2
( , ) 2 4
( ) ( )cos( ) ( )cos( )
L x i L i L i x L i x
i
(27)
With:
6
'1
0
1
() k
k
k
L i k d i
(28)
6
'1
1
1
() k
k
k
L i k e i
(29)
6
'1
2
1
() k
k
k
L i k h i
(30)
'
0()Li
,
'
1()Li
and
'
2()Li
represent the derivative of Eq
(12) (13) and (14).
The mechanical equation relating the rotor
acceleration, speed, position and load force is by Eq
(31).
2
0
2
cr
dx dx dx
m F x F signe F
dt dt
dt
(31)
Parameters
c
m
,
,
0
F
and
r
F
mean respectively the
actuator mass, the viscous friction force, the dry
friction force and the load force. The syringe force
r
F
is given by this equation.
2
2
1 2 3
2
rs a
d x dx dx dx
m C x C C
dt dt dt
dt
FF
(32)
With:
18C
,
4
3
12
2222
( ) / 8
c
t s v
K
KK
CD
DDD
,
22
2
344
8 ( )
cc
t s c
d
ts
DD
C L L L D
DD
,
d c p
L L L
,
c
D
and
2
4
cp
am
DP
FF
Parameters
v
,
,
p
P
and
designate the kinematic
viscosity, the density, the blood pressure and the
dynamic viscosity.
In order to validate the proposed accuracy model,
Malab/Simulink was used to perform the simulation
with this model. This last, has been tested and
compared by the linear model to predict the dynamic
performance of the LSRM.
Figure 7. Dynamic performances for the biomedical
system (actuator-syringe)
Dynamic behavior of displacement thrust force and
speed are resumed in figure 7. Note that, the phase A
excitation allows positioning of the translator on the
first step corresponding to 1.5 mm. Successive
excitation of other phases are needed for next steps.
0 0.5 1 1.5 2 2.5 3 3.5 4
-2
0
2
4
6x 10-3
Time (s)
Speed (m/s)
Without saturation With saturation
b- Speed
0 0.5 1 1.5 2 2.5 3 3.5 4
0
5
10
Time (s)
Thrust force (N)
Without saturation With saturation
c- Thrust force
0 0.5 1 1.5 2 2.5 3 3.5 4
-2
0
2
4
6x 10-3
Time (s)
Displacement (m)
Without saturation With saturation
a- Displacement
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.7
Imed Mahmoud, Adel Khedher
E-ISSN: 2769-2507
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Volume 5, 2023
Although it works in open loop, the system is
characterized by lack of oscillations. The rotor
movement is almost linear and there is practically no
need for a closed loop control. The saturated model is
characterized by a strongly oscillatory speed and thrust
force compared to the linear one. These oscillations are
expected to disturb the position accuracy and the speed
constancy is often required by many industrial
applications and especially in the medical fields. This
problem often leads to synchronism losses.
5. closed-loop control of the linear stepper motor
To synthesize the transfer function, it is considered that
the machine operates at a vacuum
r
F0
and the mobile
movement is ahead where
sign V 0
and
F00
.
Equation (31) becomes:
2
2,
c
dx dx
m F i x dt
dt
(33)
The transfer function between the position x and the
thrust force F is given by equation (35).
2( ) ( ) ( )
c
m s x s sx s F s
(34)
2
( ) 1
() () c
xs
Gs F s m s s
(35)
Moreover, the one phase thrust force is expressed by:
22
, sin( )F i x ki x
(36)
Where
1
L
k
.
But for each movement sequence, the actuator is
incremented by a step given by Eq (37):
4
xx
(37)
Whether:
22
sin( ) sin( ( ))
4
22
sin( ) cos( )
2
xx
xx
(38)
Since
x
is weak then
2
cos( ) 1x
From where:
2
,F i x ki
(39)
Following these developments, the block diagram of
the closed loop control of the motor can be established
by Figure.8.
Figure 8. Closed loop control of the motor
This control configuration requires two
regulators. A proportional regulator R(1) of gain kp1
for evaluating the reference speed Vref and a
proportional integral regulator R(2) gains kp2 and
integral action Ti which serves to determine the
reference force Fref. According to the expression
(39), the reference current iref can be used to control
the converter. The load force Fc developed by the
syringe is added to the system as a disturbance.
Regulator gains are determined by the pole
compensation method.
Simulation results for a time of the order of
unity infusion, figure.9 illustrates the dynamic
behavior of the assembly from control current,
position, speed and thrust force on a whole step.
In the case of a linear reference, the rotor
movement is also perfectly linear. It can be
observed that the control law functions correctly
with a trajectory tracking with a very high
precision, figure.9 (a). As expected, the current is
chopped to satisfy the motion linearity, Figure.9
(b). The average speed remains constant around
1.510-3m/s, figure.9 (c). As for the Thrust force, it
remains positive average value canceling at the end
of the step which is consistent with the operation
principle, Figure.9 (d). On the other hand, the phase
A excitation allows the positioning of the actuator
on the first equilibrium position corresponding to
1.5 millimeters. The successive excitation of the
other phases is necessary for the next positions.
LSRM
=
=
V
Ci
R(1)
R(2)
Fc
xréf
Vréf
x
V
Fréf
iréf
i
+
+
+
-
+
-
+
-
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.7
Imed Mahmoud, Adel Khedher
E-ISSN: 2769-2507
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Volume 5, 2023
The closed-loop control has therefore brought
undeniable improvements to the linear stepper
movement and the control strategy has led to
satisfactory results.
The dynamic performances of the all biomedical
system during an infusion depend on the interaction
of several parameters characterizing the volume of
infusion
infusion
V
, the syringe geometry (
118
c
L mm
and
60
c
V ml
) and the Tooth pitch of the actuator
6mm
. These parameters are related by the
following expression:
c
c
infusion
VV
L
(40)
Table 3 (appendix) shows that the infusion time is
proportional to the volume of aqueous solution to be
infused. For instance, an infusion volume of 1.5 ml
of sodium chloride corresponds to an infusion time
of four seconds which requires one power cycle of
the actuator. According to the nature of the disease
and the patient, the infusion volume is increased or
decreased and consequently the infusion time. In
this case, it is necessary to repeat or split the power
cycle of the actuator according to the need.
6. Conclusion
The magnetic materials have nonlinear
properties which are mainly illustrated by the
hysteresis phenomena and saturation. For this
purpose, this paper focuses on the modelling,
dynamic characteristics and closed loop control of a
biomedical system. To predict the motor
electromagnetic characteristics, an analytical model
of the LSRM is presented taking into account the
magnetic circuit nonlinearity. Results are compared
to those obtained via the 2D-FEM. The comparison
shows a reasonable agreement, proving the validity
of the proposed approaches. Then, dynamic
performances with and without saturation for the all
biomedical system are presented and discussed.
Finally, in order to solve the movement problem
regularity, an essential factor characterizing the
medical application. To overcome the limitations
and inadequacies of conventional control
techniques, a control concept based on closed-loop
control is developed. The application of this control
strategy made it possible to enslave the system and
force it to follow rigorously the linear reference
without overshoots and oscillations.
Figure 9. Closed loop control: Dynamic performance of the actuator for 1s infusion time
(a) : Mover displacement (b) : Control current (c) : Speed (d) : Thrust force
Control current (A)
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
Temps (s)
Courant d'alimentation (A)
(b)
Time (s)
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5 x 10-3
Temps (s)
Déplacement (m)
Position de référence
Régulation
(a)
Mover displacement (m)
Time (s)
Reference position
Regulation
Time (s)
Speed (m/s)
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2x 10-3
Temps (s)
Vitesse (m/s)
(c)
0 0.2 0.4 0.6 0.8 1
0
1
2
3
Temps (s)
Force de poussée (N)
(d)
Time (s)
Thrust force (N)
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.7
Imed Mahmoud, Adel Khedher
E-ISSN: 2769-2507
54
Volume 5, 2023
In the perspective of this work, our contribution
will be paid to the implementation of an
Experimental bench of the biomedical system for a
real validation.
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International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.7
Imed Mahmoud, Adel Khedher
E-ISSN: 2769-2507
55
Volume 5, 2023
Appendix
Table 1. Dimensional characteristics of the all biomedical
system.
Table 3. Infusion Volume versus infusion time and the
cycles number.
Table 2: coefficients of Eqs (9), (10), (11), (12), (13) and (14)
Coefficients of Eq
(9)
Coefficients of Eq
(10)
Coefficients of Eq
(11)
6
a = -0.14964
6
b = -0.29455
6
c = -0.10415
5
a = 0.40276
5
b = 1.0098
5
c = 0.39906
4
a = -0.34888
4
b = -1.2404
4
c = -0.54238
3
a = 0.18654
3
b = 0.68632
3
c = 0.31762
2
a = -0.17356
2
b = -0.22522
2
c = -0.1048
1
a = 0.0085628
1
b = 0.015891
1
c = 0.0085593
0
a = 0.15724
0
b = 0.12801
0
c = 0.099954
Coefficients of Eq
(12)
Coefficients of Eq
(13)
Coefficients of Eq
(14)
6= 0.2779d
6= -0.8095e
6= 0.3820h
5 = -1.0691d
5 = 2.9400e
5 = -1.4681h
4 = 1.4731d
4 = -3.8375e
4 = 2.0155h
3 = -0.7963d
3 = 2.0967e
3 = -1.1139h
2 = 0.0685d
2 = -0.4154e
2 = 0.1733h
1 = -0.0091d
1 = 0.0354e
1 = -0.0177h
0= 0 .1300d
0= - 0.0028e
0= 0 .0301h
Dimensions of the used syringe
volume of the syringe
V
60 ml
Piston mass
s
m
15 g
Length of the cylinder
c
L
118 mm
External diameter of the cylinder
c
D
30.11 mm
Internal diameter of the cylinder
e
D
27.48 mm
Tube length
t
L
1600 mm
Inner diameter of the tube
t
D
3.7 mm
Needle length
s
L
69.7 mm
Needle diameter
s
D
0.8 mm
Required thrust force
F
4N
Rated voltage
U
14 V
Rated current
I
1 A
Motor mechanical and electrical Parameters
Number of modules
m
4
Tooth width
b
3mm
Slot width
a
3mm
Tooth pitch
6mm
Phase separation
c
1.5mm
Mover length
Lm
135 mm
stator length
Ls
40.5 mm
Air gap width
0.1mm
Height of the stator teeth
Hs
17mm
Height of the mover teeth
Hm
4mm
Depth of the actuator
D
30mm
Number of turns per phase
520
Number of
power cycles
Infusion Volume
(ml)
Infusion time
(s)
1
1.5
4
2
3
8
3
4.5
12
4
6
16
….
….
…..
20
30
80
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
International Journal of Electrical Engineering and Computer Science
DOI: 10.37394/232027.2023.5.7
Imed Mahmoud, Adel Khedher
E-ISSN: 2769-2507
56
Volume 5, 2023
