CHINESE JOURNAL OF MECHANICAL ENGINEERING
·832·
Vol. 29,aNo. 4,a2016
DOI: 10.3901/CJME.2016.0429.063, available online at www.springerlink.com; www.cjmenet.com
Design of a Transverse-flux Permanent-magnet Linear Generator and Controller
for Use with a Free-piston Stirling Engine
ZHENG Jigui1, 2, *, HUANG Yuping2, WU Hongxing1, and ZHENG Ping1
1 Electrical Engineering & Automation Department, Harbin Institute of Technology, Harbin 150001, China
2 Beijing Research Institute of Precise Mechatronic Controls, Beijing 100076, China
Received August 4, 2015; revised April 25, 2016; accepted April 29, 2016
Abstract: Transverse-flux with high efficiency has been applied in Stirling engine and permanent magnet synchronous linear generator
system, however it is restricted for large application because of low and complex process. A novel type of cylindrical, non-overlapping,
transverse-flux, and permanent-magnet linear motor(TFPLM) is investigated, furthermore, a high power factor and less process
complexity structure research is developed. The impact of magnetic leakage factor on power factor is discussed, by using the Finite
Element Analysis(FEA) model of stirling engine and TFPLM, an optimization method for electro-magnetic design of TFPLM is
proposed based on magnetic leakage factor. The relation between power factor and structure parameter is investigated, and a structure
parameter optimization method is proposed taking power factor maximum as a goal. At last, the test bench is founded, starting
experimental and generating experimental are performed, and a good agreement of simulation and experimental is achieved. The power
factor is improved and the process complexity is decreased. This research provides the instruction to design high-power factor
permanent-magnet linear generator.
Keywords: permanent-magnet linear motor(TFPLM), stirling engine, magnetic leakage factor, power factor
1

Introduction
The combustion of fuel is used to drive an enclosed gas
in a Stirling engine for moving a piston. The free-piston
Stirling engine has a compact design and is highly efficient;
variety of fuels can be used in it and has great potential in
areas such as spacecraft, nuclear energy, solar energy, and
heat recovery[1]. In addition to their high efficiency, Stirling
engines produce little or no emissions, so this technology
can address the problems of an increasing global demand
for energy and the requirement for environmentally
sustainable methods of energy conversion. Stirling engines
require a heat source, which may be created from the
combustion of conventional fuels, such as natural gas or
petroleum, but other sources of energy, such as solar, wind,
and nuclear power, can be used[2]. Linear generator has
been becoming the current hot research topic in the field of
motor and its control for its fast response, high efficiency
and high precision[3]. The motion of a linear generator is
linear, which is also true of a Stirling engine’s piston.
Eenergy can be saved by the integration of these two
devices through greater efficiency and facilitate the
development and utilization of renewable energy.
* Corresponding author. E-mail: zhengjigui@163.com
Supported by National Natural Science Foundation of China(Grant No.
50877013)
© Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2016
Modern Stirling power systems were pioneered by the
Philips Company in the Netherland in 1938. General
Motors and Ford Motor Company in the United States,
United Stirling in Sweden, and Japan all conducted
research on Stirling systems. In the U.K., six organizations,
including Reading University, the Royal Naval College, the
University of Bath, and the Peters Company, were funded
by the British government and several financial groups to
conduct research on Stirling power systems. The Royal
Naval College designed a 1 MW marine Stirling power
system using air as the working medium. In Canada, a
group at the University of Calgary conducted a great deal
of research on the theory, testing and overall system
planning for Stirling power systems[4].
The crankshaft connecting rod system changes up and
down movable of piston into the circular motion of
flywheel in The traditional internal combustion engine
vehicles., This way of power transmission has higher
efficiency, when the load changes frequently at the good
working condition, the output power of the internal
combustion engine is far less than the rated power[5–6].
QIU S G, et al[7], provided a detailed design for each
component in a Stirling engine and a method to achieve
longer operating times without sacrificing system efficiency.
In the area of low-power systems, the Stirling Technology
Company developed the RemoteGen series of power
generation systems[8]. Maschinefabrik Augsburg-Nurnberg
(MAN) and Motorenwerke Mannheim(MWM) in Germany
CHINESE JOURNAL OF MECHANICAL ENGINEERING
and several research and manufacturing organizations in
Russia also conducted relevant research[9]. BLARIGAN, et
al[10], presented a design of a free-piston generator system
for hybrid vehicles in which a combustible gas in two
combustion chambers expanded alternately to do work and
drive a connecting rod to initiate the cyclic movement. A
connecting rod connects the engine to the linear generator’s
mover. The magnetic fields from the magnets in the mover
pass through the coils(windings) in the stator. The varying
magnetic field induces a voltage in the stator windings.
In recent years, researchers in many countries have
turned their attention to the transverse-flux, permanentmagnet linear motor(TFPLM) because of its distinct
advantages in applications, such as wind power generation
and electric vehicles. In Germany, TFPLMs are being
developed for use in electrically powered automobiles, and
TFPLMs are being introduced into maglev rail systems[11].
NOZAKI et all and KANG et all studied that injecting a
third-harmonic current into the TFPLM winding could
improve the motor’s performance[12–13]. HARRIS, et al[14],
proposed a unilateral TFPM with an external rotor that
reduced the overall diameter and volume of the motor for
the same air gap. Only when the stator magnetic field and
rotor magnetic field of TFPM are perpendicular to each
other, the analysis results of three dimensional fieldcan be
precise[15–17]. LU, et al[18], studied a type of TFPM with low
armature magnetic leakage using high-strength permanent
magnets, resulting in a higher power factor and force
density. However, the difficulty in installing the permanent
magnets may lead to demagnetization. Payne studied a
fan-shaped three-phase TFPM[19–20]. The motor had a
C-shaped stator core embedded in a disk rotor, where the
stator windings spanned 120° in the rotor for each phase.
Theoretically, the motor can have a power factor of 0.7 at
the cost of reduced torque density. Husband et al. presented
a 2 MW transverse magnetic flux switching magnetic
resistance motor for marine propulsion. This motor had a
bilateral magnetic concentrator structure, and the stator had
a C-shaped core, which provided a higher moment density.
To address the problem of relatively low power factors
exhibited by current TFPMs, this study investigated a
cylindrical, non-overlapping TFPLM. The linear motor was
designed specifically for use with a free-piston Stirling
engine to form a generator system, and the cylindrical
configuration facilitates integration with the Stirling engine.
Adjacent stator pole teeth do not overlap, so the armature
magnetic leakage is reduced and the power factor, the
air-gap magnetic flux density, and the force density is
greater. Mathematical models for the permanent-magnet
synchronous linear motor and the free-piston Stirling
engine are developed. The linear motor drives the engine in
the startup phase, and the engine drives the linear motor in
the power generation phase. A controller is developed that
uses the feedback loops of current, velocity and voltage in
the startup phase, and in the power generation phase,
different feedback loops are used for power regulation and
·833·
voltage regulation. To verify the performance of the
proposed TFPLM and the controller, computer simulations
and laboratory tests are performed.
2
Cylindrical, Non-overlapping TFPLM
Design
For the TFPLM in this study is used in a free-piston
Stirling generator system and will be integrated with an
internal combustion engine, a cylindrical configuration is
used. In this configuration, no overlap exists between to
adjacent stator poles carrying magnetic fields in opposite
directions, which reduces the armature radial magnetic
leakage and increases the power factor.
2.1 Stator structure
The stator has a circumferential, three-phase structure
with three types of laminations, as shown in Fig. 1. There
are six internal stator teeth that are evenly distributed in
Stator lamination 1, i.e., the centers of adjacent stator tooth
roots are 60° apart. Each stator tooth comprises a tooth
root and a tooth top, and the tooth tops point clockwise.
Similarly, stator lamination 2 has six stator teeth, but the
tops of the stator teeth point counterclockwise. Stator
lamination 3 also has six stator teeth evenly distributed; but
in this case, the stator teeth have only roots and no tops.
Fig. 1.
Cylindrical non-overlapping circumferential
three-phase TFPLM stator laminations
When the stator cores overlap, type 1 and type 3 stator
laminations overlap in the axial direction (i.e., the central
axis of the motor) and form a stator pole whose axial length
is one unit of the pole pitch. Type 2 and type 3 laminations
also overlap axially and form an adjacent stator pole whose
axial length is also one unit of the pole pitch. Axially, the
tooth roots of adjacent stator poles align and are tightly
arranged. Thus, the tooth tops of adjacent stator poles point
to opposite directions. Therefore, the tops of adjacent stator
pole teeth are offset from each other so that no overlap
exists.
The tooth roots for each stator pole are at the same
angular position and form a stator tooth unit. The armature
windings are wound around each stator tooth unit.
2.2 Mover structure
The mover includes permanent magnets, a core, and a
shaft. The arrangement for one phase is shown in Fig. 2.
The permanent magnets and the core are attached to the
·834·
ZHENG Jigui, et al: Design of a Transverse-flux Permanent-magnet Linear Generator and Controller
for Use with a Free-piston Stirling Engine
sides of the shaft. The permanent magnets are arranged in
groups of three to form an array, and each permanent
magnet array spans 120°. For each stator pole, there is a
permanent magnet array with three magnets. The three
permanent magnets span 30°, 60°, and 30°, in that order.
The fields of the permanent magnets are oriented
transversely, and the orientation is perpendicular to the
direction of motion. The field orientations of adjacent
permanent magnets are opposite circumferentially and
axially.
Fig. 2. Mover for the cylindrical non-overlapping TFPLM
circumferential three-phase motor
2.3 Three-phase arrangement
To obtain a three-phase motor, the permanent magnet
arrays have a sequential axial offset of 2/3 pole pitch. Fig.
3 shows the arrangement of the permanent magnets as if
they were unwound from the shaft.
the main flux Фm and the main pole magnetic leakage flux
Фmσ. The current in the armature winding generates the
armature magnetomotive force and the armature magnetic
field. The armature magnetic flux includes the armature
main flux Фa and the armature magnetic leakage flux Фaσ.
If there is no overlap between the stator core laminations
carrying an opposing armature magnetic flux, the armature
radial magnetic leakage will be reduced greatly, and thus,
the armature magnetic leakage flux Фaσ is also greatly
reduced. The main flux Фm and the armature main flux Фa
merge into the air-gap magnetic flux Ф in the gap between
the stator and the armature, and the product of the air-gap
magnetic field density and the current density is
proportional to the thrust. Therefore, adjacent stator poles
carrying opposite magnetic fluxes are non-overlapping,
which increases the air-gap magnetic flux density and the
thrust. Because of the reduction in magnetic leakage, the
leakage reactance is reduced, and thus, the power factor
increases.
The main parameters of the circumferential three-phase
motor are listed in Table 1. A complete motor has 48 poles,
whereas the simplified model only has 6 poles; therefore,
the induced electromotive force and the thrust from the
model are 1/8 those of a complete motor. The simplified
model is shown in Fig. 4.
Table 1.
Fig. 3.
Arrangement of the permanent magnets
in the three-phase TFPLM
2.4 Operation
Because of the offset and absence of overlap between
adjacent stator pole teeth, the magnetic flux generated by
the permanent magnets in the corresponding stator core
laminations has the same direction, and the magnetic fields
that pass through the windings overlap with each other.
When the mover passes, the direction of the interlinked
magnetic flux in the windings changes. Because the
distance between two adjacent phases is 2n+2/3 times the
pole pitch (where n is a non-negative integer depending on
the space filled by the end winding), the electromotive
forces, which are separated by 60° in space and phase, are
generated in the armature winding, and consequently, a
three-phase alternating current is generated.
During operation, the permanent magnets create the main
pole magnetic field. The main pole magnetic flux includes
Circumferential three-phase motor parameters
Parameter
Value
Rated power P/kW
Rated speed v/(m·s–1)
Permanent magnet material
Stator outside diameter Ds/mm
Stator yoke inner diameter ds/mm
Stator tooth outside diameter Dst/mm
Stator tooth inner diameter dst /mm
Mover outside diameter Dm/mm
Air gap length g/mm
Series windings per phase N
Poles per phase m
Pole pitch p/mm
1
3
N35SH
78
70
52
42
36
1
35
48
15
Fig. 4.
Simplified finite element model for circumferential
three-phase structure
Leakage flux of TFPLM is analysed using 3D finite
element analysis, the armature longitudinal magnetic flux
leakage is the main part of the armature magnetic flux
leakage. The reason is the distance between adjacent stator
cores less than the total length of permanent magnet and
crack, teeth overlapping area of adjacent stator iron core of
opposite direction is too large. However, there is no teeth
CHINESE JOURNAL OF MECHANICAL ENGINEERING
overlapping area between adjacent stator iron cores with
the novel type of cylindrical, non-overlapping,
transverse-flux. Non-overlapping TFPLM motor has the
advantages of easier to integrate with internal combustion
engine, less armature magnetic flux leakage and easier to
process.
3
and therefore, the magnetic leakage factor increases.
TFPLM Parameter Optimization
The magnetic leakage factor is the main index of motor
performance: the magnetic leakage factor indicates the
effectiveness of the permanent magnets, which mainly
affects the force density of a motor. The armature magnetic
leakage factor indicates the magnitude of the motor leakage
reactance, which mainly affects the power factor of a motor.
Thrust ripple is a source of motor vibration and noise,
especially during low-speed operation where resonance
occurs.
Permanent magnet is used to provide the field for
permanent magnet linear generator instead of dc supply
field coil, which makes the motor structure more simple,
processing and assembly cost is decreased, the collector
ring and brush are saved that improves the reliability of the
motor running. The efficiency of the motor is also
improved due to the fact that the excitation current is not
required. The permanent magnet of linear generator is
installed on the moving parts, and the coil is in the stator
core, using silicon steel material as the rotating permanent
magnet generator.
3.1
·835·
Effect of the core’s axial thickness
of long teeth stator
With all other dimensions of the motor fixed, including a
pole pitch of 15 mm, the core’s axial thickness of long teeth
stator was varied. The first permanent magnet was assumed
to be air, and a current was assumed in the armature
winding. Three-dimensional(3D) finite element software
was used to analyze the effect of the stator long teeth core’s
axial thickness on the armature magnetic leakage factor.
The results show that as the stator long teeth core’s axial
thickness increases, the main flux and the total flux remain
nearly the same; the percentage of the total magnetic
leakage caused by the armature transverse magnetic
leakage is far greater than that caused by the armature
radial magnetic leakage in the total magnetic leakage,
which indicates that the non-overlapping arrangement of
adjacent stator teeth significantly reduces the armature
radial magnetic leakage.
Fig. 5 shows the magnetic leakage factor versus the axial
thickness of the stator long teeth core. It can be observed
that the armature magnetic leakage factor increases
generally as the stator long teeth core’s axial thickness
increases. As the core’s axial thickness of long teeth stator
increases, the total flux, the main flux, and the armature
radial magnetic leakage are nearly constant, whereas the
armature transverse magnetic leakage increases gradually,
Fig. 5.
Armature magnetic leakage factor vs. axial thickness
of the stator long teeth core
Next, the core’s axial thickness of long teeth stator was
varied with no current in the armature winding and all other
dimensions fixed. The finite element software was used to
analyze the effect of core’s axial thickness of long teeth
stator on the performance of the permanent magnets. The
flux was nearly constant, the main flux increased initially,
then decreased, and the radial magnetic leakage of the
permanent magnets was the primary source of magnetic
leakage.
Fig. 6 shows that as the axial thickness of the stator long
teeth core increases, the magnetic leakage factor of the
permanent magnets increases initially, then decreases. The
reason is that the effectiveness of the permanent magnets is
very low when the axial thickness of the stator long teeth
core is very small, and therefore, the magnetic leakage
factor is higher. The distance between two adjacent stator
pole teeth decreases, the magnetic conductance increases,
and the permanent magnet inter-polar radial magnetic
leakage increases when the axial thicknesses of the stator
long teeth core are big enough. And thus, the magnetic
leakage factor increases.
3.2 Effect of the pole-arc coefficient
With all other dimensions fixed, the pole-arc coefficient
was varied, and its effect on the permanent magnet
magnetic leakage factor was analyzed. The results of the
3D finite element analysis showed that the total flux and
the main flux increased with the pole-arc coefficient. The
permanent magnet radial magnetic leakage was the main
source of magnetic leakage for the permanent magnets, and
the proportion decreased as the pole-arc coefficient
increased. A plot of the permanent magnet magnetic
leakage factor versus the pole-arc coefficient is shown in
Fig. 7.
Fig. 7 shows that the permanent magnet magnetic
leakage factor increases as the pole-arc coefficient
increases. The reason is that the stator long teeth core’s
axial thickness is one unit of pole pitch, so the effect of the
·836·
ZHENG Jigui, et al: Design of a Transverse-flux Permanent-magnet Linear Generator and Controller
for Use with a Free-piston Stirling Engine
axial thickness of the stator long teeth core on the
permanent magnet effectiveness is not considered. As the
pole-arc coefficient increases, the magnetic leakage
between permanent magnets increases gradually; therefore,
the magnetic leakage factor gradually increases. Based on
the air-gap magnetic density and the magnetic leakage
factor, the pole-arc coefficient was set to 12/15.
effect is generated by the change in the air-gap magnetic
conductance between the stator and the mover because of
the slot in the stator core. An expression for the cogging
force is
F=
¥
 2 zDmo æç C
2ö
çç g + g ÷÷÷ å nGn Br(nz
ø
4 è 
0
n=1
æ 2nz
ç
è C
p ) sin ç
ç
ö
x÷÷÷ ,
ø
(1)
where z is the number of stator slots, Dmo is the mover
outside diameter, μ0 is the air-gap magnetic conductivity, C
is the stator axial length, g is the air-gap distance, n is an
integer that makes nz/p an integer, Br(x) is the distribution of
the permanent magnet remanence in the air gap, and x is the
displacement of the stator in the axial direction.
Fig. 6.
Relationship between Permanent magnet magnetic
leakage factor and other variables
3.3 Thrust ripple analysis
The main cause of thrust ripple is the existence of the
detent force. The detent force is the force generated by the
interaction between the primary core and the secondary
permanent magnets in the axial direction. There are two
types of detent forces: one generated by the interaction
between the ends of the stator core and the permanent
magnets, called the end effect, and the other generated by
the interaction between the stator teeth and the permanent
magnets, called the cogging force. In the cylindrical,
non-overlapping TFPLM, the end effect is a special
phenomenon caused by the stator core with a finite length.
For the stator core is open at the end, It leads to an abrupt
change in the air-gap magnetic resistance at the end of the
core, thereby causing a cyclic ripple in the thrust.
In the cylindrical, non-overlapping TFPLM, the cogging
Fig. 7.
Relationship between thrust ripple and other variables
Eq. (1) shows that Br(x) has a significant effect on the
cogging force. However, not all coefficients in the Fourier
decomposition of Br(x) will affect the cogging force; only
the (nz/p)th Fourier coefficients affect the cogging force.
The cogging force can be reduced by adjusting the pole
pitch of the stator and the mover. Within the range of a
single tooth slot, the relative positions of the stator and the
mover change, so the cogging force also changes
periodically, where the number of cycles is determined by
matching the numbers of poles and slots. The number of
CHINESE JOURNAL OF MECHANICAL ENGINEERING
cycles is the minimum integer n for which nz/p is an
integer. Within the range of a single tooth slot, a greater
number of cogging force cycles results in a smaller cogging
force. However, in the cylindrical non-overlapping TFPLM,
the pole pitches of the stator and the mover are equal, so
the number of poles and slots cannot be changed to reduce
the cogging force.
The pole-arc coefficient also affects the coefficient of the
Fourier decomposition. Therefore, the cogging force can be
reduced by selecting values of the pole-arc coefficient that
result in smaller values of the Fourier coefficient.
A plot of the thrust ripple versus the stator long teeth
core’s axial thickness is shown in Fig. 7(a). The graph
shows that when the stator long teeth core’s axial thickness
is 10 mm, the thrust ripple reaches a minimum value of
7.37%.
A plot of the thrust ripple versus the pole-arc coefficient
is shown Fig. 7(b), which reveals that the thrust ripple
reaches a minimum value when the pole-arc coefficient is
equal to 0.8.
3.4 Thrust density and power factor analysis
The power factor and the force density are the main
indexes of concern for the design of the cylindrical
non-overlapping TFPLM in this study. The relationships
between the motor parameters and the power factor and the
force density are essential to the optimization of the
parameters. The power equation is
Fv = mEph I ph cos  ,
(2)
where F is the thrust of the TFPLM, v is the motor speed, m
is the number of motor phases, Eph is the induction phase
potential, Iph is the phase current, and cosφ is the power
factor.
The motor induction phase potential Eph is defined as
Eph = 2fNpm ,
(3)
where N is the number of windings per phase, p is the
number of poles, and Фm is the magnetic flux per pole.
The magnetic flux per pole is defined as
m = Bδ
Dsi
ls ,
4
(4)
where Bδ is the air-gap magnetic density.
The winding current density J is defined as
J=
I ph
Sc
,
(5)
where Sc is the cross-sectional area of one turn of the
conductor.
·837·
The motor thrust can be expressed as
F=
l
3 22
NpBδ J s Dsi Sc cos  ,
8

(6)
and the motor force density is
Fξ =
l D S cos 
F
3 2
=
NJBδ s si c 2
.
2
æ D ö÷

2
Do 
ç
o ÷
p çç
÷
èç 4 ø÷
(7)
The power factor can be defined as
cos  =
Eph
2
2
Eph
+ ( LI ph )
,
(8)
.
(9)
where ω is the electrical angle.
If Id =0, then
cos  =
Eph
2
2
Eph
+ ( LI q )
Substituting from the previous equations gives
1
cos  =
2
1+
32 ( LI q )
.
(10)
2
æ
l ö
4 çç NpBδ vDsi s ÷÷÷
çè
ø
Eqs. (7) and (10) show that the motor force density and
the power factor are related to ls/  , which is the pole-arc
coefficient.
When the pole-arc coefficient is fixed, the axial thickness
of the stator long teeth core regulates the force density by
changing the effectiveness of the permanent magnets. The
stator long teeth core’s axial thickness regulates the power
factor by changing the armature magnetic leakage.
Fig. 8 shows that when the pole-arc coefficient increases,
the power factor increases initially and then decreases. The
force density and the power factor have similar trends. The
main reason is that for lower values of the pole-arc
coefficient, the air-gap main flux increases with the
pole-arc coefficient, and consequently, the power factor and
the force density increase. For larger values of the pole-arc
coefficient, the space between adjacent permanent magnets
decreases and the magnetic conductance increases, so the
permanent magnet radial magnetic leakage increases
sharply and the air-gap main flux decreases, leading to a
decrease of the power factor and the force density.
Fig. 9 shows that the power factor gradually declines as
the stator long teeth core’s axial thickness increases, and
the force density increases initially and then decreases as
·838·
ZHENG Jigui, et al: Design of a Transverse-flux Permanent-magnet Linear Generator and Controller
for Use with a Free-piston Stirling Engine
the stator long teeth core’s axial increases thickness. When
the axial thickness of the stator long teeth core is 10 mm,
the power factor reaches a maximum value.
Fig. 10.
Controller architecture with voltage
and current feedbacks
4.1
Fig. 8.
Power factor and thrust density vs. pole-arc coefficient
Mathematical model of the free-piston
stirling engine
The power produced by the free-piston Stirling engine is
determined from the average pressure, the piston stroke
length and the temperature ratio of the hot and cold sections.
The power relationship is shown in
æ 1ö
P µ Pmean X p2 çç1- ÷÷÷ ,
çè  ø
(11)
where P is the average output power, Pmean is the average
pressure, Xp is the piston stroke length, and τ is temperature
ratio of the hot and cold sections. Eq. (11) shows that the
output power increases with the temperature ratio τ and the
stroke length. The power equation is
P = KFPSE X p2 ,
Fig. 9. Power factor and thrust density vs. axial
thickness of the stator long teeth core
(12)
where KFPSE is engine power coefficient, given by
4
System Simulation and Test
The engine can operate normally only when the piston in
the free-piston Stirling engine is at a stable oscillating
frequency and reaches a certain stroke length. An external
force is required to drive the piston when the engine starts
from rest. The linear motor provides this force in the startup
phase. A storage battery provides the power for the linear
motor. The mover drives the piston through a connecting
rod.
In the startup phase, the spatial vector method is used to
control the motor, and the controller uses current, speed,
and position feedback loops. When the Stirling engine
reaches normal operating conditions, the engine drives the
linear motor, which functions as a generator, and the
electrical energy that is generated is stored in the storage
battery. When the power generated is approximately equal
to the load, the controller uses voltage and current feedback
loops, as shown in Fig. 10. The rectified direct current (DC)
output voltage is maintained in the proper range using the
storage battery voltage as feedback, and the output either
charges the storage battery or provides electricity for other
devices.
æ 1ö
K FPSE = KPmean çç1- ÷÷÷ ,
çè  ø
(13)
where K is the power ratio coefficient.
In an actual system, Pmean and τ are fixed by the design,
but the stroke length Xp can be varied so that the output
power matches the load. The free-piston Stirling engine can
be viewed as an equivalent mass-spring-damper system, for
which the thrust equation is
æ Xp
ö
F = k ççç
- x÷÷÷ ,
çè 2
ø÷
(14)
where x is the piston displacement (with respect to one end
of the stroke) and k is an equivalent damping coefficient.
4.2
Mathematical model of the cylindrical
non-overlapping TFPLM
It is assumed that there is no damping effect in the
permanent magnets and no damping in the mover windings
and that the magnetic field is sinusoidal across the air gap.
In addition, core saturation and vortex and hysteresis losses
CHINESE JOURNAL OF MECHANICAL ENGINEERING
are ignored. Given these assumptions, the voltage equations
of the linear motor in the d-q coordinate system are
ìï
ïïud = d d +  v q + rs id ，
ïï

dt
ïï

d

ïï
q
+ v d + rs iq，
íuq =
ïï

dt
ïï
d 0
ïï
ïïu0 = dt + rs i0，
ïî
from the voltage equation, the equation of motion, and the
displacement equation as follows:
ìï did

ïï Ld
= ud + v d - rs id ，
ïï dt

ïï
d
i

ïï
q
= uq + v q - rs iq，
ïï Lq
t

d
ïï
ïï di
0
= u0 - rs i0，
í L0
ïï dt
ïï
ïïM dv = F - F - B v，
e
l
v
ïï dt
ïï
ïï dx
ïï = v.
ïïî dt
(15)
where the magnetic linkages d, q, and 0 are defined as
ìï d = Ld id + f ，
ïï
ï = L i ，
í q
q q
ïï
ïï = L i .
0 0
î 0
(20)
(16)
4.3
Using the relations
ìï
ïïns = f ，
ïï
p
ïï
ïï
r
ïï = ，
p
ïï
ïïv =  r = 2n r，
ïí
s
ïï x = vt，
ïï
ïï
x
ïï = ，
ïï

ïï
ïï P = 3 p  i - i v = F v，
( d q q d)
e
ïïî
2
·839·
(17)
Simulation of system startup and power
generation
In the startup phase, the free-piston Stirling engine
requires stable piston motion. A storage battery provides
electricity for the linear motor. The linear motor drives the
Stirling engine until the stroke length reaches the required
value. Position, speed, and current feedback loops are used
in the control system with id=0. Pulse width modulation is
used to control the spatial voltage vector pulse. The
maximum stroke length is set to 14 mm. The engine power
coefficient is KFPSE=5.1´106, and the equivalent
impedance coefficient is k=200 kN/m. The system is
simulated with commanded stroke lengths of 10 mm and 14
mm. Stable motion is achieved in both cases. Fig. 11 shows
the results of the simulations for a half cycle of the piston
in the two cases.
the following expressions are obtained:
ì
v = 2 f ，
ï
ï
ï
ï
v
ï
ï
= ，
ï
í

ï
ï
ï
3
ï
Fe = p ( d iq - q id )，
ï
ï
2
ï
î
(18)
where v is speed of the traveling wave in the magnetic field,
p is the number of pairs of linear motor poles, f is the
frequency of the stator armature current, τ is the pole pitch,
x is the mover displacement, θ is the spatial vector angle,
and Fe is the electromagnetic thrust.
The equation of motion of the mover is
M
dv
= Fe - Fl - Bv v ,
dt
(19)
where M is the mass of the mover, Fl is the load resistance,
and Bv is a mechanical damping coefficient.
Using the preceding equations and the stator currents ia,
ib, ic, the speed v, and the displacement x as state variables,
the state equations of the linear motor can then be derived
Fig. 11. Simulation results for 10 mm and 14 mm commanded
stroke lengths, trapezoidal waves with fixed speed
In Fig. 11, the horizontal axis is time in seconds. The
subfigures from top to bottom are the phase currents ia, ib, ic
(A), the commanded speed v*(m/s), the actual speed v
·840·
ZHENG Jigui, et al: Design of a Transverse-flux Permanent-magnet Linear Generator and Controller
for Use with a Free-piston Stirling Engine
(m/s), and the displacement s(m).
The simulation results show that the displacement
reaches the commanded stroke length without any
overshoot for the given frequency. The speed tracked the
commanded speed profile closely. Through the entire stroke,
the speed varied smoothly; the speed curve declines
smoothly to 0 at both ends of the stroke, so the system
startup will be stable.
When the Stirling engine reaches normal operating
conditions, the system switches to the power generation
phase. In this phase, the Stirling engine drives the mover of
the linear motor to generate power. The controller uses
voltage and current feedback loops to maintain the voltage
at the desired value. A three-phase rectifier bridge module
and a resistance load are connected to measure the DC bus
voltage and current to calculate the motor output power.
The system was simulated with the value of the
resistance load R in the three-phase rectifier bridge module
set to 20 Ω and 60 Ω. The simulation results are shown in
Fig. 12, where the horizontal axis is time(s), the upper
graphs show the DC bus voltage Vdc(V) and the lower
graphs show the output power P(W).
Fig. 12. Simulation results for voltage regulation
with 20 Ω and 60 Ω load resistances
The results show that the voltage feedback loop ensures
that the DC bus voltage is relatively stable in the power
generation phase. Although there is no position control loop
because the speed changes much more slowly than the
voltage, the stroke is not significantly affected and the
output power is relatively stable. Overall, the system
responses for both loads are relatively stable.
4.4 Laboratory tests
Fig. 13 shows a diagram of the control system. The
controller consists of two main parts: the driver module
(DM) and the control module(CM). The DM includes a
power drive and isolation circuit, a detection and isolation
circuit, and a protection circuit. The control module
includes a TMS320F2812 DSP controller(Texas
Instruments Inc., Dallas, TX, USA) and peripherals, a
position detector, and a fault handling circuit.
At the system level, the DM drives the linear motor and
provides protection for the linear motor. In addition, the
DM measures the three-phase alternating current and the
bus voltage and current, digitizes the analog signals and
passes them to the DSP controller for subsequent
processing. The CM is responsible for measuring the mover
position, processing the motor voltage and current signals
in real time, and performing the vector control calculations.
In addition, the CM provides an interface with a personal
computer(PC). Fig. 14 shows a photograph of the actual
controller.
Fig. 13. Diagram of the control system for the free-piston
Stirling engine and the permanent-magnet linear generator
Fig. 14. TFPLM system Controller
In this test, a brushless DC motor was used to simulate
the free-piston Stirling engine. A crankshaft and a
connecting rod were used to convert between rotary and
linear motion, as shown in Fig. 15.
Fig. 15. TFPLM system test platform
With the loops of position, speed and current closed,
stroke control in the startup phase begins. The mover
frequency was set at 12 Hz, and commanded stroke values
of 8 mm and 12 mm were tested. Time histories of the
mover position for the two commanded stroke values are
CHINESE JOURNAL OF MECHANICAL ENGINEERING
shown in Fig. 16.
·841·
(3) Nowadays the transverse flux permanent magnet
motor(TFPM) has low power factor and complex structure
technology, this problem is solved by the proposed new
structure.
(4) The controller prototype based on DSP is developed,
and the simulation results show the power voltage under
different loads with less volatile and better robustness.
References
Fig. 16.
Mover position vs. time for a commanded stroke length
The current-loop test shows that the controller works
well and can perform PWM control. The graph of mover
position shows that the stroke length of the mover
increased to the commanded value gradually at a constant
frequency.
In the power generation test, voltage regulation in the
power generation phase was tested. The voltage and current
loops were closed, and mover speed was set to 0.5 m/s.
The voltage set point was 15 V, and the DC side of the
system was connected to a 12 V storage battery. Test results
of the output voltage for 6 Ω and 9 Ω load resistances are
shown in Fig. 17, respectively.
Fig. 17.
DC voltage vs. time with different load resistance
The test results show that the output voltage has
relatively low ripple for both loads and the system
robustness is good when the voltage is being regulated(in
the power generation phase).
5 Conclusions
(1) The new structure of transverse flux permanent
magnet linear motor can meet the free piston Stirling
generator system at low speed, big thrust requirements.
(2) Efficiency is raised and the environmental protection
is easier with the help of the structure of transverse flux
permanent magnet linear motor.
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Biographical notes
ZHENG Jigui, male, born in 1979, is currently a doctoral
candidate at Electrical Engineering & Automation Department,
Harbin Institute of Technology, Harbin, China. His research
interests include designing of complex electromechanical control
system and the drive transmission and actuation of Servo.
Tel: +86-13671077411; E-mail: zhengjigui@163.com
HUANG Yuping, born in 1967, is currently a general researcher at
Beijing Research Institute of Precise Mechatronic Controls, China.
His research interests include complex electromechanical control
system, making breakthrough on some of the key and core
technologies.
Tel: +86-10-88530861; E-mail: huangyp@2008.sina.com
WU Hongxing, male, born in 1975, is currently an professor at
Harbin Institute of Technology, China. His research interests
include permanent electric machines and control, switched
reluctance motor and control, and unconventional electromagnetic
devices.
Tel: +86-451-86403086; E-mail: whx0422@sina.com
ZHENG Ping, femal, born in 1962, is currently an professor at
Harbin Institute of Technology, China. His research interests
include electric machines and control, hybrid electric vehicles,
and unconventional electromagnetic devices.
Tel: +86-451-86416048; E-mail: zhengping@hit.edu.cn