Design, Modeling and Analysis of Linear Switched Reluctance Motor for Ground Transit Applications

IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE)
e-ISSN: 2278-1676,p-ISSN: 2320-3331, Volume 10, Issue 1 Ver. III (Jan – Feb. 2015), PP 01-10
www.iosrjournals.org
DOI: 10.9790/1676-10130110 www.iosrjournals.org 1 | Page
Design, Modeling and Analysis of Linear Switched Reluctance
Motor for Ground Transit Applications
1
Praveen Kumar C, 2
K Geetha, 3
Madhavi K
Abstract: Translations are the most common motions in several industrial branches. Usually the linear motion
is generated by belt or spindle drives using rotational motors. Elasticity and backlash are the problems that
hinder the accuracy of such drives. These mechanical arrangements provide low speed only. Utilization of high
performance linear motors can eliminate these problems because they generate linear motion without any extra
mechanical components. Linear switched reluctance motor (LSRM) is one of the linear motors that can be used
in such drives. The design and mathematical modeling of a linear switched reluctance motor suitable for ground
transit applications is presented in this paper.
Key words: LSRM, SRM, switched reluctance, flux linkage
I. Introduction
The development of linear actuators is today a research field of interest to the field of engineering.
Usually rotating to linear motion conversion is obtained through mechanical gears. Since this process is
completely mechanical, a special maintenance is a must if high reliability and operational ability are demanded.
Other drawbacks such as space or efficiency can be eliminated with the introduction of linear drives. Research is
now focused on the invention of new materials and technologies with the ability to directly produce a linear
displacement and, simultaneously, is also trying to solve issues related with linear actuators like the cogging
force, the attractive force perpendicular to the motion [1]. A standard design procedure for a single-sided and
longitudinal-flux-based linear switched reluctance machine (LSRM) is developed in [2], [5]. The proposed
design procedure utilizes the rotating switched reluctance machine (rotary SRM) design by converting the
specifications of the linear machine into those of an equivalent rotating machine. The machine design is carried
out in the rotary domain, which then is transformed back into the linear domain. Linear machine drives
combined with electromagnetic levitation are best suitable for conveyor applications in semiconductor
fabrication plants and in low- and high speed transit applications because of their ability to produce propulsion
force on the rotating part, known as the translator, without mechanical contact and friction. Linear switched
reluctance machines (LSRMs) are the counterparts of the rotating switched reluctance machines. In fact, the
linear switched reluctance machine is obtained from its rotary counterpart by cutting, along the shaft over its
radius, both the stator and rotor and then rolling them out. A linear SRM may have windings either on the stator
or translator (the moving part), whereas in the rotary switched reluctance machine the windings are always on
the stator and the rotor contains no windings. The fixed part is called either a stator or track and the moving part
is called a translator. There are two distinct configurations of linear SRM in the literature: longitudinal flux and
transverse flux. These two configurations can be obtained by unrolling both the stator and rotor of a rotary SRM
with a radial magnetic flux path and axial magnetic flux path, respectively. The converter topology with a
minimum number of power devices and control implementations to facilitate the pulsation- free force control of
the linear switched reluctance machines is investigated in [4]. Design of a high speed linear switched reluctance
motor (LSRM) is presented in [6]. In [7], four longitudinal LSRM configurations are presented and designed to
operate as propulsion actuators in a vertical elevator prototype. The most suitable LSRM configuration based on
simplest construction, highest payload capability, and low copper loss, which is the modified high force density
LSRM, is analyzed, designed, implemented, and tested as a propulsion actuator in a vertical elevator prototype.
In [8], a self tuning regulator (STR) based on the pole placement algorithm is proposed for high-precision
position tracking of the LSRM. A novel magnetic levitated switched reluctance linear actuator is proposed in
[9]. The actuator is designed for applications in small to medium-scale automation machinery. The linear
actuator has a simple and robust structure and is suitable for high-speed and high-precision position control with
direct drive capability.

Design, Modeling and Analysis of Linear Switched Reluctance Motor for Ground Transit Applications
(a) Longitudinal Flux Path Configuration
(b) Transverse Flux Path Configuration
Fig. 1 Three-Phase Linear LSRM with Active Stator and Passive Translator
II. Principle Of Operation
Linear switched reluctance machines (LSRMs) can be good alternative to linear induction or synchronous
machines because of lack of windings on either the stator or rotor structure. Some of the features of these motors
are:
• Lack of windings on either the stator or rotor structure.
• Absence of mechanical gears.
• Ideal for manufacturing and maintenance as the winding are concentrated.
• Windings are connected in series with a switch, so that the inductance of the winding can limit the rate of
change of rising current, in case of a shoot through fault.
• The phases are independent and provide uninterrupted operation of the motor drives in the event of failure
of one winding.
• Absence of significant heat sources during secondary operation and only one part of the secondary that is
opposite to the primary is present in the magnetic field.
Some of the applications of LSRM are:
 Material handling systems.
 Transporting materials inside a totally enclosed system.
 Food processing plants, to move the items from one place to another.
A reluctance motor is an electric motor in which torque is produced by the tendency of its movable part
to align to a position where the inductance of the excited winding is maximized. A switched reluctance motor
(SRM) has simple construction compared to induction or synchronous machines. The structure of the switched
reluctance motor constitutes a stator and a rotor as in the case of any other motor. The stator is composed of
steel laminations shaped to form poles. The windings are always on the stator and rotor has no windings as
shown in Fig.2. But a linear SRM may have windings either on the stator or translator. Longitudinal flux and
transverse flux configurations for four-phase LSRM with an active translator and passive stator structure are

shown in Fig.1. The longitudinal magnetic flux path configuration is a linear counterpart of three-phase radial
flux rotary SRM. The flux path in this machine is in the direction of the vehicle motion (Fig.1.a). A transverse
flux design (Fig.1.b) has the flux path perpendicular to the direction of vehicle motion. It allows a simple track
consisting of individually mounted transverse bars. As the flux is perpendicular to the direction of motion, an
emf is induced in the core, resulting in high eddy current losses. If the stator is active and translator is passive,
SRM configuration has the advantage of having the stationary power supply and power converters, resulting in
reduced weight of the vehicle. But this design requires a large number of power converter sections along the
track, resulting in high costs. When a pair of stator windings that are connected in series is excited, the translator
tends to move so as to align itself with the magnetic flux axis of the excited stator phase windings. This position
corresponds to the fully aligned position and has the maximum phase inductance. The position corresponding to
maximum reluctance value and hence minimum phase inductance is called the unaligned position and occurs
when a corresponding pair of translator poles that eventually will be aligned is half a translator pole pitch away
from the axis of the excited stator poles. The translator moves forward smoothly when the stator windings are
switched in sequence. According to the converter topology and the mode of operation, the previously excited
phase may be turned off before or after the succeeding phase is excited. Reverse motion of the translator is
achieved by reversing the excitation sequence of the stator phases.
The operation of the linear SRM depends on the inductance profile of the machine. The inductance of
the machine depends on the machine dimensions such as the stator and translator pole and slot widths, excitation
currents and translator position. The inductance characteristics are independent of stator current excitation since
the magnetic circuit is assumed to be linear and using this, a relationship between the machine dimensions and
inductance, shown in Fig.3, is derived.
Fig. 2 Three Phase Rotary SRM
Fig. 3 Inductance Profile and Generation of Force of LSRM
The inductance of a phase winding is its self-inductance.
Five translator positions are significant to derive the inductance profile and they are given by,

x1 =
Wts −Wsp
2
(1)
x2 = x1 + Wsp
x2 =
Wts +Wsp
2
(2)
x3 = x2 + Wtp − Wsp
= Wtp +
Wts −Wsp
2
(3) x4 = x3 + Wsp
= Wtp +
Wts +Wsp
2
(4)
x5 = x4 +
Wts −Wsp
2
(5)
= Wtp + Wts (6)
Where Wtp is the width of translator pole, Wts is the width of the translator slot, Wsp is the width of
the stator pole, and Wss is the width of the stator slot. Between x2 and x3, there is complete overlap between
stator and translator poles, and inductance during this interval is a maximum which corresponds to the aligned
value. Zero propulsion force is generated in this region since there is no change in the inductance in this region.
But this flat inductance region is important to give time to commutate the current and hence to prevent the
generation of a negative force. Flat-top inductance profile is due to the unequal stator and translator pole widths.
The regions corresponding to 0 to x1and x4 to x5 have no overlap between stator and translator poles. These
positions have the minimum phase inductance called as unaligned inductance. The rate of change of inductance
is zero and hence these regions do not contribute to propulsion force production. The forward direction of
motion of the translator is assumed to be positive when the phase excitation sequence is abc. For reverse
direction of motion, the phase sequence is acb. One half of the back emf power is stored in the form of magnetic
field energy in phase windings and the other half of back emf power (or air gap power) is converted to
mechanical power output.
III. Modeling Of LSRM
Current is generated by the magnetic field in an electromagnet.
A. Energy Analysis
We know that the instantaneous voltage across the terminals of a single-phase SRM winding is related to the
flux linked in the winding by
Faraday’s law,
v = iR +
dφ
dt
(7)
Where
v is the terminal voltage
i is the phase current
R is the motor resistance
φ is the flux linked by the winding
The flux linkage in a SRM varies as a function of rotor position, θ and the motor current i.
Thus the equation can be represented as:
v = iR +
dφ
di
di
dt
+
dφ
dθ
dθ
dt
(8)
Where
dφ
di
is defined as winding inductance L (θ, i), which is a function of rotor position and current.
Multiplying each side of equation by the electrical current, i, gives an expression for the instantaneous power in
an SRM.
vi = i2
R + i
dφ
dt
(9)
= i2
R +
dφ
di
di
dt
(10)
The left hand side of the equation represents the electrical power, delivered to the SRM. The first term on the
right hand side represents the ohmic losses and the second term represents the electric power at coil terminal,
which is a sum of mechanical output and any power stored in SRM.
Pe = L
di
dt
I (11)
The relation between power and energy is,
dWe
dt
= Pe (12)

Where We is a part of the total energy delivered to the winding, which is a sum of energy stored in the coil Wf
and energy converted into mechanical work, Wm . It can be written as:
We = Wf + Wm (13)
The magnetic field energy Wf can be given by the equation,
Wf = idφ
φ
0
(14)
The graphical interpretation of the field energy is shown in the Fig.5. The energy stored in the magnetic field
with the air gap g, can be expressed in terms of magnetic flux density Bg as follows,
Wf =
Bg
μo
dBgVg=
Bg
2μo
Vg (15)
Fig. 4 Graphical Interpretation of Magnetic Field Energy
The field energy is inversely proportional to permeability and directly proportional to volume of air gap.
The area below the curve in Fig.5 is defined as magnetic Co- field energy. Co- field energy Wf′ is defined by
Wf
′
= φdi
i
0
(16)
If φ − i is non linear then Wf′ >Wf. If φ − i characteristic is linear then Wf′ = Wf.
If the secondary part has moved slowly the current, i= v/R remains the same at both positions in the steady state
because the coil resistance does not change and the voltage is set to be constant.
Fig. 5 Illustration to magnetic force derivation
The operating point has moved upward from point a to b (Fig.6).
B. Calculation of Force
The forces in the x, z, and y directions, named as the propulsion force, normal force, and lateral force,
respectively, and shown in Fig. 6 is calculated using the principle of conservation of energy expressed as,
dWi = dWl + dWs + dWm (17)

where dWi is the incremental electric input energy, dWl is the incremental loss energy dissipated by heat, dWs is
the incremental stored magnetic energy, and dWm is the incremental mechanical energy. Assuming that the
translator moves in x direction by da, the incremental mechanical energy is defined as,
dWma = fa × da (18)
where fa is the propulsion force generated. The same expression is used if appropriate
Fig. 6 Analytical Force Calculation Using Conservation of Energy
Direction and the displacements in that direction is for any other force. It is clear from the conservation
of energy principle that the force calculations in x, z, and y directions require incremental values of electric
input energy, loss energy dissipated by heat, and stored energy in the magnetic field.
The flux density in the air gap is given by,
∅ =
Bg
A B
(19)
where A and B are the overlapped lengths in a particular position of the translator with respect to the stator in
the x and y directions, respectively. Assuming a large air gap in the linear machine structure, Equation 19 can be
written as,
∅ = Bg. AB = μ0Hg. AB = μ0
Tph i
C
A B (20)
= Tph i =
∅ Z
μ0 A B
(21)
dWe = i dλ = i d Tph ∅ = Tph i d∅ =
∅ Z
μ0 A B
(22)
The stored energy in the magnetic field Ws is,
Ws =
Bg
2
A B C
2 μ0
=
∅
A B
2
.
A B C
2μ0
=
∅2C
2μ0A B
(23)
dWsa = −
∅2 C
2μ0 A2B
da +
∅ C
μ0 A B
d∅ (24)
Ignoring incremental energy due to losses such as copper loss, core loss, and mechanical friction loss, the
incremental mechanical energy dWma during the incremental displacement da can be written as,
dWma = dWi − dWsa
=
∅ C
μ0A B
d∅ +
∅2
C
2μ0 A2B
da −
∅ C
μ0 A B
d∅
=
∅2C
2μ0 A2B
da (25)
From which the propulsion force is now obtained as,
fa =
dWma
da
=
∅2C
2μ0 A2B
=
BgA B
2
C
2μ0 A2B
=
Bg
2
2μ0
A B (26)
The incremental stored magnetic energy dWsc during the incremental displacement dc, which is the cause of the
normal force, is given by,
dWsc =
∅2C
2μ0 A2B
dc +
∅C
2μ0 A2B
d∅ (27)
Ignoring incremental energy due to losses, the incremental mechanical energy dWmc during the incremental
displacement dz can be written as,

dWmc = dWi − dWsc
=
∅ C
μ0 A B
d∅ −
∅2
2μ0 AB
dc −
∅ C
μ0 A B
d∅
= −
∅2
2μ0 AB
dc (28)
The normal force is then obtained as,
fc =
dWmc
dc
= −
∅2
2μ0 AB
= −
BgA B
2
2μ0 AB
= −
Bg
2
2μ0
A B (29) fb =
Bg
2
2μ0
A B
(30)
In the prototype design described in with Lw = 57.75 mm, Wsp = 23.1 mm, g = 1 mm, and Bg = 1.122 T, the
propulsion force due to excitation of a single pole is,
fa1 =
Bg
2
2μ0
BC (31)
=
Bg
2
2μ0
× Lw g (32)
= 28.901 N
The propulsion force due to the excitation of a phase is fa= 2×fa1 = 57.802 N.
The normal force due to excitation of a single pole is calculated as,
fc1 = −
Bg
2
2μ0
A B (33)
= −
Bg
2
2μ0
Wsp Lw
= -668.205 N
The total downward force experienced by the translator is given by fc = 2×fc1 = −1336.41 N.
The normal force can be calculated at different translator positions by changing the overlap length A.
Flux linkages in the machine are calculated for a given excitation and a translator position using
magnetic equivalent circuit. The flux linkages can be obtained from the inductance and corresponding excitation
current.
The magnetic flux path of an LSRM involves five parts of the machine, namely, air gaps, stator poles,
stator yoke, translator poles, and translator yoke. Each part experiences different flux densities and different
lengths of flux line based on the chosen magnetic flux path. The mmf applied to a phase winding at any
translator position is given by equation 34.
Fig. 7 Lumped Parameter Model of Equivalent Magnetic Circuit of LSRM
FP = F1 + F2 = Fag + Fst + Ftl (34)
where FP is the total mmf per phase applied, Fag, Fst, and Ftl are the mmf drops in the air gap, stator iron core,
and translator iron core, respectively.
The mmf equation can be rewritten in terms of the magnetic field intensity and flux path length using Ampere’s
Circuital Law as,
FP = Tph i = Σ Hag lag +Σ Hst lst +Σ Htl ltl (35)

where Hag, Hst, Htl, lag, lst, and ltl are the magnetic field intensities and flux path lengths in the air gap, stator iron
core, and translator iron core, respectively.
Fig. 8 Velocity and Required Force Profiles of LSRM
IV. Design Of A Prototype With Active Stator And Passive Translator
An LSRM prototype id designed for a length of 6 m, with a maximum linear velocity of 3 m/s and acceleration
time of 1 second. The maximum mass of translator assembly is restricted to 20 kg.
Acceleration is given by,
aa =
Vm
ta
= 3 m/s2
(36)
Force for initial acceleration is calculated as
Fa = Mtaa = 60 N (37)
The deceleration, ad = −3m/s2
and deceleration force Fd = - 60 N
Power capacity of LSRM is
P = Fa × Vm = 180 W (38)
The RSRM is assumed to have a stator pole angle
βs= 30° = 0.524 rads
Rotor pole angle βr= 36°= 0.628 rads
Ke= 0.4, Kd= 1, K2= 0.7, Bg= 1.1215T
Asp= 23886.5
K= 0.655
Bore diameter, D=
pπ
60 Ke Kd K1 K2 Bg Asp Vm
(39)
= 88.17 mm
Stack length of RSRM is
L= KD (40)
= 57.75 mm
Stator yoke thickness, Csy is given by
Csy =
D βs
2
= 23.1mm (41)
Assuming D0= 190 mm, height of stator pole hs =
D0
2
−
D
2
− Csy = 27.815 mm (42)
Rotor back iron width Cry and the height of the rotor pole (translator pole) hr are then calculated as,
Cry =
D
2
βr = 27.69 mm (43)
hr =
D
2
− g − Cry = 15.4 mm (44)
Magnetic field intensity in the air gap is calculated as,
Hg =
Bg
μ0
= 892461.3A/m (45)
For a peak phase current of Ip= 8.5 A,
Number of turns per phase is
Tph =
Hg 2g
Ip
= 210 turns/phase (46)
Assuming a current density of J= 6 A/mm2
,

Area of conductor is,
ac =
Ip
J m
= 0.818 mm2
(47)
The closest wire size for this area of cross section of the conductor is AWG # 18 and it has an area of 0.817mm2
and it is selected for the phase windings.
A. Conversion from RSRM to LSRM
The number of sectors of the LSRM and the resultant total number of stator poles are
Nsc =
Lt
πD
= 22 (48)
n = NsNsc = 132 (49)
Width of stator pole and width of stator slot are given by,
Wsp =
D βs
2
= 23.1mm (50)
Wss =
πD−6Wsp
6
= 23.1 mm (51)
Translator pole width and translator slot width are,
Wtp = Cry = 27.7 mm
Wts =
πD−4Wtp
4
= 41.55 mm (52)
Total length of the translator is given by,
Ltr = 6WtP + 5Wts = 373.95 mm (53)
The core stack width of the LSRM is obtained from stator stack length of RSRM as,
Lw = L = KD = 57.75 mm
Diameter of the conductor is given by,
dc =
4ac
π
= 1.02 mm
Assuming width of wedges, w= 3 and packing factor, ff = 0.8, number of vertical layers of winding and number
of horizontal layers of winding are obtained as,
Nv = ff
hs − w
dc
= 20 (54)
Nh =
Tph
2 Nv
= 7 (55)
Stator winding area is given by,
2 ac Nv Nh
ff
= 286.3 mm2
(56)
Fill factor is calculated as
ff =
stator winding area
stator slot window area
= 0.499 (57)
Fill factor is in the range 0.2<= FF < 0.7
Finally, the following condition is to be satisfied.
Ns Wsp + Wss = Nt Wtp + Wts (58)
The duty cycle of a winding in the three-phase LSRM, 1/(3Nsc) whereas that of the RSRM winding it is 1/3.
Therefore, the windings in the LSRM can have much lower copper volume but taking more losses.
V. Conclusion
A novel design procedure for linear switched reluctance machine was proposed using the current
knowledge and design procedure of rotating switched reluctance machine. The force analysis of the prototype
was also carried out. The principle of operation of the linear switched reluctance motor was described along
with modeling.
Acknowledgement
We sincerely acknowledge with the financial support extended by Centre for Engineering Research and
Development (CERD), Thiruvananthapuram for the execution of this project work.
About The Authors
Mr Praveen Kumar C is working as Assistant Professor in the Department of Electrical and Electronics
Engineering at NSS College of Engineering, Palakkad, Kerala. He received the ME Degree in Mechatronics
from Anna University in 2013. His areas of interest include Special Machines, Linear Machines and Jet
propulsion.

Dr K Geetha is working as Professor in the Department of Electrical and Electronics Engineering at NSS
College of Engineering, Palakkad, Kerala. She has around 32 years of teaching experience in various fields
related to electrical and electronics. Her research interests include power electronics and drives and special
machines.
Ms Madhavi K is pursuing her M Tech degree in Power Electronics at NSS College of
Engineering, Palakkad. She is doing her Master Research Project in the area of Linear Switched
Reluctance Machines.
References
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Design, Modeling and Analysis of Linear Switched Reluctance Motor for Ground Transit Applications