IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 12, DECEMBER 2008
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Modeling and Control of Solid-Rotor Synchronous Reluctance Machines
Based on Rotor Flux Dynamics
Jae-Do Park1 , Claude Kalev1 , and Heath Hofmann2
Pentadyne Power Corporation, Chatsworth, CA 91311 USA
The Pennsylvania State University, University Park, PA 16802 USA
We present a model suitable for use in a vector control algorithm for synchronous reluctance machines with solid conducting rotors. The model takes the rotor flux dynamics into consideration. It is similar to an induction machine model, yet includes a magnetic
saliency of the rotor. Here, we discuss techniques for parameter extraction and suggest a modification of the model to incorporate the
nonlinear magnetic phenomena. We also investigate the influence of nonlinear magnetics on the model-based controller. Our model
yields improved performance for a fast-changing torque command compared to the conventional model when utilized in a current regulator. Experimental results on a 120 kW, 55 000 rpm machine in a flywheel energy storage system validate the performance of the
proposed model.
Index Terms—Flux dynamics, nonlinear modeling, remanent magnetization, solid rotor, synchronous reluctance motor.
NOMENCLATURE
I. INTRODUCTION
Stator flux-linkage vector in synchronous
reference frame.
Rotor flux-linkage vector in synchronous
reference frame.
Stator current vector in synchronous reference
frame.
Rotor current vector in synchronous reference
frame.
Stator inductance matrix in synchronous reference
frame.
Rotor inductance matrix in synchronous reference
frame.
Mutual inductance matrix in synchronous
reference frame.
Rotation matrix
.
Stator voltage vector in synchronous reference
frame.
Rotor resistance matrix.
Stator resistance.
Electrical rotor angular velocity.
Mechanical rotor angular velocity.
Pole number of the machine.
Stator leakage inductance.
Electrical rotor position.
Digital Object Identifier 10.1109/TMAG.2008.2003501
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
HE synchronous reluctance machine has received renewed
attention with the development of field-oriented control
theory and power electronics technology. A singly-excited synchronous reluctance machine can be a relatively simple, lowcost configuration compared with other types of machines due
to the nonexistence of windings or permanent magnets on the
rotor. Especially, it has advantages in certain high-speed applications such as flywheel energy storage systems [1]. This machine has zero “spinning” losses when no torque is being generated by the machine, as opposed to permanent magnet machines
with a stator iron.
The rotors of synchronous reluctance machines tend to consist of two different types: conventionally (i.e., transversely)
laminated, and axially laminated. In the case of the conventionally laminated rotor, the saliency of the machine is achieved
by punching flux barriers into the laminations. As a result, the
thick, flux-carrying portions of the rotor are connected through
thin “ribs.” The resulting lamination therefore has poor structural characteristics, and cannot handle extremely high-speed
applications where the centrifugal forces are large. Thickening
the ribs to improve structural performance results in a reduction
of the saliency of the rotor, and therefore reduces the electrical
performance.
Axially-laminated rotors consist of layers, or laminations, of
(soft) magnetic and nonmagnetic materials where the nonmagnetic layers provide the requisite flux barriers discussed in the
prequel. In most cases the nonmagnetic material is also an electrically insulating material, in order to impede the flow of eddy
currents in the rotor. However, such a structure is also inappropriate for use in extremely high-speed applications due to the
relatively poor bond strengths that can be achieved between the
magnetic steel and the electrical insulator, as well as the relatively poor strength of insulators themselves.
Both of the rotor designs discussed above also suffer from a
relatively high effective modulus of elasticity due to their laminated nature. This also complicates high-speed operation, as a
mechanical resonance associated with the rotor structure could
fall within the speed range of the application. If the nonmagnetic
T
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layers of an axially laminated rotor consist of a metal, however,
then the magnetic and nonmagnetic layers can be bonded together through a high-strength bonding process, such as brazing.
Such a rotor, referred to as a “solid” rotor in this paper, possesses high strength and a low modulus of elasticity, and therefore high mechanical resonance frequencies. However, the solid
rotor does not have any impediments to the flow of eddy currents. These eddy currents generate heat in the rotor, and so great
care must be taken in the design of the machine and the electric
drive to minimize these losses [1], [2]. Furthermore, as will be
discussed in great detail in the sequel, these eddy currents can
significantly affect the dynamic characteristics of the machine.
The rotor currents in synchronous reluctance machines
have been omitted in recent equivalent-circuit-based models
[3]–[6]. Therefore, existing models for synchronous reluctance
machines are inadequate if the machine has a solid rotor, as it
does not account for the resulting flux-linkage dynamics associated with a conducting rotor. In particular, when attempting a
torque step from zero to full torque, the error associated with
neglecting the rotor flux dynamics is significant, as the rate
of change of the flux linkage is determined by the rotor time
constants. The conventional synchronous reluctance machine
model can therefore create a current overshoot during transients, as the predicted back-emf is much higher than the actual
back-emf of the machine.
In this paper, we present a model for synchronous reluctance
machines with solid conducting rotors. The developed model
takes the rotor flux-linkage dynamics into consideration, which
are similar to those of an induction machine model yet include
a magnetic saliency of the rotor. First, the dynamic model of
a solid-rotor synchronous reluctance machine is presented.
Techniques for parameter extraction are then discussed. Based
upon the proposed model, a current regulator is developed and
implemented.
A modification of the model and controller are also suggested to incorporate the nonlinear magnetic phenomena of
the machine. The influence of nonlinear magnetics on the
model-based controller is investigated. The suggested approach
can be applicable to compensate magnetic saturation and
remanent magnetization.
The proposed model yields an improved performance for a
fast-changing torque command compared to the conventional
model when utilized in a current regulator. Experimental results
of such a system are presented and discussed.
II. FULL-ORDER MODEL OF SYNCHRONOUS RELUCTANCE
MACHINE WITH ROTOR FLUX DYNAMICS
IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 12, DECEMBER 2008
Fig. 1. Equivalent circuit in synchronous reference frame model of synchronous reluctance machine.
The voltage equations for the machine in the synchronous
reference frame can be obtained as follows:
(1)
(2)
where
(3)
(4)
(5)
(6)
and
(7)
The and
denote vector and diagonal matrix notation in
arbitrary reference frame. The subscripts d and q represent direct
and quadrature values, respectively. The superscript r represents
the synchronous reference frame.
The rotor currents cannot be measured, hence we represent
the stator flux linkages (3) in terms of stator currents from (4)
(8)
(9)
The voltage equations can therefore be rewritten as
(10)
A. Synchronous Reference Frame Model
Although an electrically conducting solid rotor of a synchronous reluctance machine is technically a continuum system
[7], it can be simply modeled in the synchronous reference
frame through conceptual, shorted direct, and quadrature
windings on the rotor, similar to what is typically done with
squirrel-cage induction machines. Fig. 1 presents an equivalent
circuit model of a synchronous reluctance machine in the
synchronous reference frame.
(11)
By defining a new vector
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PARK et al.: MODELING AND CONTROL OF SOLID-ROTOR SYNCHRONOUS RELUCTANCE MACHINES
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case, the stator voltage of the machine will be due solely
to the flux generated by rotor currents
TABLE I
SYNCHRONOUS RELUCTANCE MACHINE PARAMETERS
(17)
From voltage measurements we can therefore easily determine the flux-linkage . From the exponential decays of the
voltage waveforms we can also estimate the rotor time con. This can best be done through a curve fitting
stants
of the measured data. We can determine the rotor excitation re,
sistances from the conditions at the turn-off transition
for both direct and quadrature axes, as follows:
Equations (10) and (11) can be given as
(13)
(14)
We will choose the states of the system to be the vectors
and . Hence, the machine dynamics can then be written as
follows:
(18)
The resulting parameters of a four-pole, 120 kW, 55 000 rpm
solid-rotor synchronous reluctance machine are shown in
Table I.
C. Effect of Solid Rotor on Machine Torque
(15)
The effect of the solid-rotor can be seen in the torque expression of the machine. The co-energy of a three-phase solid-rotor
synchronous reluctance is derived as follows:
(16)
With this formulation, the dynamics can be expressed in terms
of three sets of direct and quadrature parameters, and a scalar
parameter:
;
• rotor time constants
• rotor “excitation” resistance
;
;
• “leakage” inductance
• stator resistance .
(19)
(20)
The electromagnetic torque is therefore given by
(21)
B. Parameter Extraction
are
Both the direct and quadrature values of
approximately equal to the stator leakage inductance
, and
,
hence can be estimated, as well as the stator resistance
through terminal measurements of the stator with the rotor reand
can be demoved. The parameters
termined from voltage and current measurements using the following procedure.
• Using a feedback current regulator at medium speeds, comto the mamand either a direct or quadrature current
chine, where the subscript x stands for the direct or quadrature component.
• Instantaneously turn off all transistors in the 3-phase inverter driving the machine at time
. The stator current
should quickly (ideally instantaneously) go to zero. In this
(22)
Under steady-state conditions, the rotor currents are zero, and
the torque expression returns to its usual form
(23)
D. Model-Based Controller
To validate the proposed model, a model-based controller has
been designed to determine the appropriate command voltages
to be applied to the machine for a desired current. The controller has been applied to a synchronous reluctance machine
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 12, DECEMBER 2008
Fig. 3. Nonlinear magnetic behavior of direct- and quadrature-axis flux
linkages.
Fig. 2. Model-based current control algorithm in synchronous reference frame.
in a flywheel energy storage system. Because of the nature of
a flywheel energy storage system (i.e., slowly changing rotor
speed), it is straightforward to model the machine dynamics accurately, and hence a model-based controller can be effective. A
model-based controller can be an attractive approach also for a
synchronous reluctance machine, where the voltage is a strong
function of current.
A simple, yet sufficiently accurate stator voltage command
for a desired stator current can be obtained by neglecting
derivative terms in (14)
(24)
is determined from the
The estimated flux-linkage vector
desired stator current vector by numerically integrating the following differential equations:
(25)
A schematic of this controller is shown in Fig. 2.
the skin effect and proximity effect will cause the effective
rotor resistance to increase with frequency. The rotor resistance
will also be a function of the rotor temperature, which in turn
will depend heavily upon the magnitude of the rotor currents.
However, it is impractical to incorporate all of these phenomena
into a single model, especially for the purposes of control.
For the model-based feedforward controller proposed in
Section II-D, it is critical to estimate flux linkage precisely
for accurate command voltage synthesis. The modeling of the
nonlinear magnetic behavior becomes more important when
an observer or an open-loop controller is adopted. Assuming
linear magnetics for all operating conditions will deteriorate
control performance, especially in the high- or low-end of the
current range, due to flux saturation or remanent magnetization,
respectively. A conceptual graph in Fig. 3 shows the nonlinear
relationship between current and flux linkage with magnetic
saturation and remanent magnetization.
In earlier works regarding the synchronous reluctance machine [6], [8], an ideal model, which did not take the nonlinear
magnetics into account, was considered. More recent studies
have focused on the issues of magnetic saturation [3]–[5]. As
can be seen in Fig. 3, magnetic saturation decrease the flux
linkage/current ratio considerably and consequently affects the
performance of the machine.
When a linear relationship between current and flux linkage
is supposed and iron loss and other second-order effects are disregarded, the steady-state flux-linkage expression is given as
follows:
III. MODEL IMPROVEMENT CONSIDERING
NONLINEAR MAGNETICS
(26)
A. Modeling of Nonlinear Components
The model of the synchronous reluctance machine in the
previous section is based on linear magnetic behavior, assuming
that the flux linkages in the machine are linearly proportional
to the currents. However, practical machines do not behave
linearly, and their nonlinear phenomena cannot be disregarded
when dealing with the control of real machines. There are
several nonlinear phenomena that make the linear modeling
difficult, such as saturation, hysteresis, stator iron loss, or
cross-coupling. The rotor resistance is nonlinear as well, since
However, when nonlinear magnetics are considered, the fluxlinkage expressions are functions of both direct- and quadratureaxis stator currents
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(27)
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PARK et al.: MODELING AND CONTROL OF SOLID-ROTOR SYNCHRONOUS RELUCTANCE MACHINES
Practically, the cross-coupling effect can be neglected in the
normal load range. Thus, the stator flux linkage/current relationship in the controller can therefore be modeled with current-dependent inductances
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straightforward to measure from the machine terminals, the inaccurate flux-linkage estimation due to the nonlinear magnetics
will become a major source of the current tracking error.
C. Incorporating Nonlinear Magnetics Into
the Model-Based Controller
(29)
B. Effect of Nonlinear Magnetics on Current Regulation
As well as the inaccurate operating point issue, a model-based
controller will experience a current tracking problem if the magnetic nonlinearity is not considered, because the flux-linkage estimation will become inaccurate.
The voltage equation of the synchronous reluctance machine
has been given as (13). Assuming a steady-state condition, the
machine terminal voltage will be
(30)
Note that the rotor flux dynamics have been neglected here because of the steady-state assumption. The proposed feedforward
controller generates the command voltage based on (30). Hence,
values which are varying due to the nonlinear phethe
nomena, will impose an effect on current regulation because the
voltage is a function of the estimated flux.
The voltage command equation is determined as follows, including errors in the inductance values
(31)
(32)
Hence, the error of steady-state voltage command will be given
as
(33)
Note that stator resistance is assumed to be accurately measured.
The voltage command, including error term, is applied to the
machine and the voltage/current relationship at the machine terminal yields as follows from (30):
(34)
Hence, the current regulation error will become
From the simplified voltage (30), flux linkages can be expressed as follows. They can be experimentally obtained by applying a series of voltages on one axis while zero voltage is applied to the other
(36)
(37)
Figs. 4 and 5 show the experimentally measured
and
of the synchronous reluctance machine that is utilized in
the experiment in Section IV. Applied voltages have the range
V and
V, and the resulting currents
A and
A for
and
have the range
measurement, respectively. The command voltages are used
to calculated the inductance values, and the rotational speed is
approximately 35 000 rpm.
The synchronous reluctance machine under study has not
shown significant magnetic saturation in the tested range, due to
its relatively large air gap. However, a remanent magnetization
of the iron in the rotor of the machine is present, as can be
seen in Figs. 4 and 5. The effect of remanent magnetization
in a synchronous reluctance machine has not been a focus of
research. Since it is the remaining flux in the magnetic circuit
when the external excitation is reduced to zero, it has generally
been a topic for sensors or small motors. However, this phenomenon can also cause an error in the low current range for a
model-based-controller driven machine. It is especially true if
the rotor material’s coercive force is not low enough to neglect
the effect of remanent magnetization.
In the linear-magnetic-model-based controller, the flux
linkage was estimated by (25). This linearly estimated flux
linkage has been compared with values determined from the
experimentally measured data points in Figs. 4 and 5. Although
the direct- and quadrature-axis flux linkage can be estimated
quite well with a single inductance value for the higher current
range, the differences are not negligible in the lower current range due to the remanent magnetization. Therefore, the
flux-linkage estimator should be modified to incorporate the
nonlinearity.
The rotor leakage inductance can be neglected because of the
fact that the mutual inductance is much greater. Hence, (12) can
be approximated as follows:
(35)
(38)
As can be seen in (35), the current tracking error is proportional
to the inductance deviation. Considering the stator resistance is
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 12, DECEMBER 2008
Fig. 4. (a) Experimentally measured direct-axis flux linkage as function of current. (b) Approximate linear flux linkage/current relationship.
Fig. 6.
mated.
L
curve: (a) experimentally measured (data points x) and (b) esti-
Fig. 5. (a) Experimentally measured quadrature-axis flux linkage as function
of current. (b) Approximate linear flux linkage/current relationship.
Fig. 7. L
curve: (a) experimentally measured (data points x) and
(b) estimated.
The rotor voltage equation can be rewritten as
(40)
where
(41)
(45)
(42)
The flux-linkage estimator can therefore be given as
(43)
(44)
can be obtained
The nonlinear flux-linkage equations
from the experimentally measured flux linkages, which can be
seen in Figs. 6 and 7. The nonlinear flux-linkage estimator (43)
and (44) can then be utilized for voltage command synthesis
(24), instead of linear flux-linkage estimator (25). The equivalent circuit is shown in Fig. 8.
Although the rotor leakage inductance is neglected to simplify the expression, this should result in little error, since the
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PARK et al.: MODELING AND CONTROL OF SOLID-ROTOR SYNCHRONOUS RELUCTANCE MACHINES
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Fig. 8. Equivalent circuit model with a nonlinear flux linkage/current relationship. Boxed inductance is nonlinear.
Fig. 10. Experimental setup.
Fig. 9. Four-pole synchronous reluctance rotor and flywheel rim.
inductance is dominated by the mutual inductance for most machines. Also, this modification does not require any additional
parameter measurement. Since the proposed model already
takes the flux dynamics into consideration, this modification
can model the flux behavior in the solid-rotor synchronous reluctance machine accurately with a straightforward procedure.
Fig. 11. Experiment: Direct and quadrature axis current regulation. Model does
not include the rotor flux dynamics. 400 A peak current command at 35 000
rpm. Experiment is at minimum-current operating point of machine. i ; i
(a) command and (b) actual (from top).
IV. EXPERIMENTAL VALIDATION
The proposed model and controller were validated on a
120 kW, 55 000 rpm, 4-pole synchronous reluctance machine.
The controller has been implemented in a digital signal processor (DSP) board [9]. The machine is part of a flywheel
energy storage system manufactured by Pentadyne Power
Corporation. The rotor consists of alternating layers of a ferromagnetic and nonmagnetic material. A picture of the machine
rotor and flywheel rim is shown in Fig. 9. The block diagram
of experimental system is shown in Fig. 10.
A. Linear Magnetics Model
Figs. 11 and 12 show the direct and quadrature current during
a 400 A peak current step command using the controller based
upon the linear-magnetics-model without rotor flux dynamics,
and the equivalent response using a linear-magnetics-model
based controller where the rotor flux dynamics has been
included. The machine is running at the minimum current
operating point, hence the synchronous reference frame current
commands are 282.84 A for both axes. The actual synchronous
reference frame currents in the figures are converted in the
controller from the measured stationary reference frame cur-
rents. The approximate rotor speed during these experiments
was 35 000 rpm. Current overshoots can be clearly seen in
the case where the rotor flux dynamics are neglected, while
they are successfully suppressed by the proposed model-based
controller.
B. Nonlinear Magnetics Model
The proposed modification has been validated on the same
experimental setup. If nonlinear magnetics are not taken into account in the controller, current regulation at low current levels
is erroneous, as can be seen in Fig. 13, since the linear-magnetics-model-based controller with a fixed inductance fails to
generate appropriate voltage command for the current range of
0–250 A. This agrees well with the variation of flux linkage in
that current range in Figs. 4 and 5. The machine currents track
the command well when the command is increased to around
250 A, because the controller parameters have been determined
accurately in that range.
Since it was difficult to fit the entire flux linkage/current relationships with a single polynomial equation, piecewise equations in Table II have been utilized, and Figs. 6 and 7 show
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 12, DECEMBER 2008
TABLE II
PIECEWISE FLUX-LINKAGE EQUATIONS
Fig. 12. Experiment: Direct and quadrature axis current regulation. Model includes the rotor flux dynamics. 400 A peak current command at 35 000 rpm.
Experiment is at minimum-current operating point of machine. i ; i (a) command and (b) actual (from top).
Fig. 13. Experiment: 0–300 A ramp commands in rotor reference frame at
35 000 rpm. Linear-model-based controller. Upper: direct-axis, lower: quadrature-axis. (a) Command current ~i and (b) actual current i .
the experimentally measured and estimated flux linkages. Although the results in Fig. 14 are not perfect because the estimated flux-linkage curves that were utilized in the experiment
are not precise enough to perfectly fit the measured data, the
result shows that the current error of the 0–250 A range is reduced and the remanent magnetization phenomena can be compensated by the proposed modification of the model. A better
tracking performance can be expected with more accurate parameter measurement and estimation.
V. CONCLUSION
A model for synchronous reluctance machines with solid
conducting rotors has been proposed. It has been shown that
the machine can be modeled more accurately if the rotor
flux-linkage dynamics associated with the solid conducting
rotor are included. Provided the model parameters agree well
with the actual system, good performance can be achieved.
A modification of the solid-rotor synchronous reluctance machine model to incorporate the nonlinear magnetic phenomena
has also been suggested. It has been validated that the controller based on the proposed modified model can remove the
current tracking error caused by the remanent magnetization.
The suggested approach can apparently be applicable to compensate magnetic saturation with extended nonlinear equations
including the saturated range.
Fig. 14. Experiment: 0–300 A ramp commands in rotor reference frame
at 35 000 rpm. Nonlinear-model-based controller. Upper: direct-axis, lower:
quadrature-axis. (a) Command current ~i and (b) actual current i .
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Manuscript received January 07, 2007; revised July 22, 2008. Current version
published January 08, 2009. Corresponding author: J.-D. Park (e-mail: jaedo.
park@pentadyne.com).
Jae-Do Park received the B.S. and M.S. degrees in electrical engineering from
the Hanyang University, Seoul, Korea, in 1992 and 1994, respectively, and the
Ph.D. degree from the Pennsylvania State University, University Park, in 2007.
From 1994 to 2001, he was a research engineer with LG Industrial Systems,
Anyang, Korea. Since 2004, he has been with Pentadyne Power Corporation,
Chatsworth, CA, where he is currently a Controls Software Engineer. His current interests include flywheel energy storage systems, power converters, and ac
machine drives.
Claude Kalev earned his undergraduate degree in electronic engineering from
the California Polytechnic State University, San Luis Obispo.
He joined Pentadyne Power Corporation in June of 2002 as Vice President,
Electrical Engineering. He was a co-founder of Pentadyne when the company
was incorporated in 1998. His interests are high-speed rotating machinery, magnetic bearing system development, high vacuum systems, and molecular drag
pump design.
Mr. Kalev is a member of Tau Beta Pi and Golden Key Honor Society.
Heath Hofmann received the B.S. degree in electrical engineering from the
University of Texas at Austin in 1992. He received the M.S. in 1997 and Ph.D.
in 1998, both in electrical engineering and computer science from the University
of California at Berkeley.
He is an Associate Professor at Pennsylvania State University, University
Park. His research area is power electronics, specializing in the design and control of electromechanical systems. Specific interests are energy harvesting (i.e.,
the generation of electricity from one’s environment), flywheel energy storage
systems, and electric drives for electric and hybrid electric vehicles.
Dr. Hofmann was awarded a Prize Paper Award (First Prize) by the Electric
Machines Committee at an IEEE Industry Applications Society Annual Meeting
in 1998.
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