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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS,
VOL.
42 NO. I , FEBRUARY
1995
17
E
A
Maximum Torque Control with a Controlled
Capacitor for a Single-phase Induction Motor
Tian-Hua Liu, Member, ZEEE
Abstract-This paper presents a new method to achieve a maxi-
mum torque for a single-phase inductionmotor. An ac adjustable
capacitor using an electronic switch in parallel with a capacitor
is proposed. The capacitor is short-circuited n a different period
by an electronic switch during each cycle to vary the effective
value of the ac capacitor.Two new optimization algorithms, which
obtain a maximum starting torque by adjusting the effective
capacitor, are proposed. No starting capacitor
or
centrifugal
switch is used here. A theoretical analysis, and simulated and
experimental results are presented in this paper.
I. INTRODUCTION
SINGLE-PHASE induction machine
( S P I M )
is widely
A
sed in industry as well as in households. In a typical
home, more SPIMs are used than any other kind of machine
111-[3]. The
SPIM,
due to its structure and its capacity for
mass production, is rugged and cheap. Many papers have
proposed different control algorithms to improve the perfor-
mance of a single-phase induction motor [
11-[6].
However,
these papers needed either
SPIMs
of special design or a
complex drive circuit. The most popular types of SPIM in
industrial applicatioqs are installed with two capacitors. One
is used during the starting period to increase the starting torque,
and the other is used during the running condition to improve
efficiency [7]. However, the use of two capacitors has some
disadvantages. For example, the cost of starting and running
capacitors combined is almost double the cost of a single
capacitor, and it requires a centrifugal switch to disconnect
the starting capacitor as the motor runs near its operating
speed. Typical applications for such motors are compressors,
pumps, air condensers, and other pieces of equipment that
must start under a load. The function of the capacitor is
to generate another leading phase from the supply voltage
source to feed an auxiliary winding so that the motor can
be operated as
a
balanced two-phase machine. For this reason,
the capacitor value must be carefully chosen according to the
terminal impedance of the auxiliary winding. Unfortunately,
this impedance changes dramatically from the starting phase
to the running phase. If both the largest starting torque and
the best running conditions are needed, at least two capacitors
must be used with the auxiliary winding. The motors with two
capacitors are called capacitor-starting and capacitor-running
motors. An electronically controlled capacitor, which uses just
Manuscript received July 17, 1993; revised June
25,
1994. This research
received support from the National Science Council of the Republic
of
China,
under
Grant
NSC-83-0416-E-011-001.
The author is with the Department of Electrical Engineering, National
Taiwan Institute of Technology, Taipei, Taiwan 106, Republic of China.
IEEE
Log
Number
9407735.
one capacitor with electronic switch to improve the perfor-
mance of a single-phase induction motor, has been proposed
by T. A . Lip0 et al.
181-[9];
however, these published papers
did not study how to obtain a maximum torque for
S P I M s .
This paper studies two methods to obtain the maximum torque
of the SPIM and then proposes a realizable system which uses
a small running cap acitor in parallel with one solid-state switch
as shown in Fig. 1.By suitably adjusting the duty cycle of the
solid-state switch, a maximum starting torque can be achieved.
The system consists of three major elements: an induction
machine, a switched capacitor and a digital signal processor
DSP). The small running capacitor required for optimum
running conditions is permanently connected in series with
the auxiliary winding, and a bidirectional switch is connected
in parallel with the capacitor. The main winding of the
SPIM
is directly connected to the main power supply. During the
starting period, the capacitor is short-circuited periodically
in order to increase the effective value of the ac capacitor.
The maximum possible starting torque can be achieved by
suitably adjusting the switch on and off during each cycle and
changing the length of the shorting interval. Unlike previous
papers [1]-[6], this paper focuses on using a very simple
circuit configuration to achieve a maximum starting torque
for the SPIM. The other distinctive feature in this paper is
the implementation of an on-line adjustment of the switched
capacitor by
a
DSP and the validation of the development
algorithms by computer simulation and experimental results.
The paper is organized as follows. A mathematical model
for the single-phase induction motor is given in Section 11.
The mathematical model is established by using d q
equations in the stationary reference frame, and then a steady-
state equivalent circuit is derived. The principle of a switched
capacitor and two optimal methods are proposed in Section
111. Using the first method, an on-line torque computation
is proposed, and then the shorting period of the switched
capacitor to obtain a maximum torque can be determined.
Using the second method, an optimal capacitor to produce
a maximum torque is derived, and then the shorting period
of the switched capacitor is obtained. Section
V I
provides
both simulated and experimental results. Finally, concluding
remarks are given in Section
V.
11.
MATHEMATICAL ODEL
Since the stator windings of the
SPIM
are not identical,
i t
is necessary to transform the variables of the
SPIM
into a
stationary reference frame in order to obtain voltage equations
with constant parameters. The equivalent SPIM circuit, in
02784046/95 04.00
1995 IEEE
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 42, NO. I , FEBRUARY 1995
Aux . Winding
I
control
t u r n - o n p e r i o d ?
DSP
Fig.
1.
Block diagram of the proposed system.
terms of a stationary reference-frame is shown in Fig. 2(a)
and (b). It
is
customary to refer the rotor variables to the
stator windings by turn ratio. The mathematical model of the
SPIM can be expressed as the following equations [7]:
where,
V;, and V:s qs and d -axis stator voltages,
R,, and
R d
4 - and $ -axis resistances,
P differential operator d / d t ,
wb
synchronous angular speed,
X,, and Xds 4'- and ds-axis stator self reactances,
i and i;, 4'- and d s-axis stator currents,
X,, and Xmd qs and d -axis magnetizing reactances,
4 - and d -axis rotor currents,
Nq and Nd
qs
and #-axis effective turns,
WT
rotor angular speed,
Rhr and R , 4 -and ds-axis rotor resistances, and
X,
and
Xir
4 -
nd @-axis rotor self reactances.
The 4 - nd d -axis self reactances can be expressed as
a'
r and izr
Xqs
=
lqs + %my
x d s
=
Xlds + Xmd
X:r = XIqr+Xmq
XAr = X / d r x m d (8)
5 )
6)
7)
I -
G-
\
R s J x i W
C
Fig.
2.
transient
state. (b)
qs-axis transient
state.
(c) Steady-state operation.
Equivalent circuits for a single-phase induction motor. (a) d -axis
where
Xlqs
and
Xlds
are the
4 -
and @-axis stator leakage
reactances, Xiqr and
XIdr
re the ys- and @-axis rotor leakage
reactances.
The instantaneous electromagnetic torque can be expressed
as
T , =
~ / 2 ) N ~ / N q ) x m q / W b ) i ~ s 2 ~ r
i : r )
(9)
where
P
is the number
of
poles for the motor.
The electromech anical equation of the machine is
10)
where J , is the inertia constant of motor and load, and
T l
is
the extemal load.
From Fig. 1 , the d -axis winding
is
the auxiliary winding,
and the q -axis winding is the main winding. It is obvious
1
p w, =
-(Te Z
Jm
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LIU: A MAXIMUM TORQUE CONTROL WITH A CONTRO LLED CAPACITOR FOR A SINGLE-PHASE INDUCTION MOMR
~
19
that VqS is equal
to I SOUrCer
and
where V,o,,,e is the input voltage source of the motor, C is
the capacitor connected in series with the auxiliary winding.
Assuming the voltages and currents of the motor are sinu-
soidal, and letting
p
= w , and then substituting into 1)- 4),
the steady-state equivalent circuit can be obtained and shown
as Fig. 2(c)
[7],
where
v:s+
s the positive sequence voltage,
GS-s the negative sequence voltage, lis+ s the positive
sequence current,
I
is the negative sequence current, s is
the slip of the induction motor. A ccordin g to Fig. 2(c), the
performance of the SPIM is not only affected by the parameters
of the motor, but also influenced by the capacitor connected
with the auxiliary winding.
111.
CONTROL
STRATEGY
A . Principle
of
Switched Capacitor
The switched capacitor circuit, which is shown in Fig.
1,
consists of two p arts: an ac capacitor and an electronic switch.
The switch must allow bidirectional current flow as in a
mechanical switch. The switch is short-circuited and opened
for each cycle. It is closed the instant the voltage across
the capacitor reaches zero; thus a zero voltage switching is
performed. Before the switch is closed, the current flows
through the capacitor which is placed in series with the
auxiliary winding. When the switch is short-circuited, the
current bypasses the capacitor and the voltage across the
capacitor is zero. As a result, the voltage across the capacitor
may change only during the period of the switch being opened.
By closing and opening the switch periodically, the effective
value of the capacitor appears to be larger than the actual
value
[9].
When the short-circuited period of the switch is
longer, the voltage across the capacitor is lower; however,
the current flowing through the auxiliary winding is increased.
Therefore, for a fixed frequency,
it
is obvious that the effective
capacitor is increased when the shorting period of the switch
is longer. So, it is possible to obtain a larger capacitor to start
the induction motor by using just a small capacitor (running
capacitor) in parallel with a switch. The shorting period
y
of the switch can
be
determined by optimization methods
implemented by a DSP.
B. Optimization Methods
The object to be optimized can
be
one of the following
quantities: power factor, input power, efficiency, torque, or
input current of main circuit. In this paper, a maximum starting
torque of an induction motor is proposed. From (1)-( 1 I , it is
possible to generate a contour graph of torque as a function
of the speed and capacitor. A typical contour graph is shown
in Fig.
3.
The significant characteristic of this figure is that
the maximum SPIM torque is a function of both the speed
and capacitor. Moreover, the capacitor must
be
adjusted at
different speeds
in
order to obtain
a
maximum torque. It
is clear that one can choose an optimal path to control the
0 40
Fig. 3.
Contour graph of torque to capacitor and speed.
capacitor to achieve a maximum torque (see Fig. 3). As can
be se en from the figure, the significa nce is that the torque of the
SPIM reaches its maximum value at a larger capacitor when
the motor is at standstill. After the motor starts, the capacitor
related to the maximum torque begins to decrease. However, it
is very difficult to directly derive the SPIM maximum torque
and its related capacitor from the differential (l )- (l l), because
they are five order, nonlinear, couple differential equations. In
addition, the measurement of the rotor currents itr and i ,
is very difficult in the real world. In this paper, a method
of symmetrical components to describe the variables of the
SPIM as positively and negatively rotating components shown
in Fig. 2(c) is used. The torque-speed curve derived from a
steady-state equivalent circuit is very close to the average
torque-speed curve obtained from a transient state. This will
be shown in the latter section of this paper.
Method 1: The fundam ental components of the stator cur-
rents are obtained by computing Fourier Series of the ds-$
axis currents. Then an average torque can be calculated from
the equivalent circuit. The fundamental components of the
stator currents are expressed as
3 r T f 2
3 e--jnw e
. s
,, t)dt
where
I
is the fundamental component of the q-axis current,
I
is the fundamental component of d-axis current, T is the
period of the current waveforms, and t is time. Then the
positive and neg ative sequence stator currents are
where
I
is the q-axis equivalent positive sequence stator
current, L t s - is the q-axis equivalent negative sequence stator
current. The positive and negative sequ ence rotor currents can
be derived from Fig. 2(c), and expressed as
(16)
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20 EEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL.
42
NO. 1 , FEBRUARY 1995
I = 3Xmq s (17) The maximum torque at different speeds is achieved by letting
dTe/dXc
=
0
and then from
(20),
the condition
of
the
maximum torque can be derived as the following equation:
( jxmq+ X{qr)
+
( R k r / ( 2
s))+-
-qr-
where
gr+
s
the q-axis equivalent positive sequence rotor
current,
I
is the q-axis equivalent negative sequence rotor
current. The average torque is
C s ( M ?
+
N?)(RaS, R b Q 1 )
+
{M1(-2Xlqs Xa b ) +Nl(2Rqs+ Ra Rb)}.
- (L -)- Rr
]
18)
x {Rb(P?
Q:)
Ra(T:
5:))=
0 (29)
2 - s
we
In this paper, a maximum starting torque control algorithm
is proposed. For every slip, there exists just one shorting
interval, which can produce a maximum torque. The shape of
the maximum torque curve formed by measuring is an arc,
so
there exists a feasible direction to reach the maximum torque.
By using the steepest ascent method
[lo],
the shorting period
to reach the maximum torque can be obtained as
where
PI,
Q1,
SI
I , re dummy variables and expressed as
From (29), it is easy to understand that the capacitor reactance
needed to achieve the maximum torque varies with the speed
and the parameters of the motor. A 1-hp single-phase induction
motor, the parameters of which are shown in Table I, is used
in the paper. Based on the parameters, an optimal capacitor
curve achieving maximum torque is shown in Fig. 5.From
this curve, it can be seen that the starting torque reaches its
maximum value for a given capacitor when the motor is at
standstill. After this point the capacitor related to the m aximum
torque begins to decrease as the speed increases. However, the
optimal starting torque, which depends on the speed and the
parameters of the motor, can be represented by a nonlinear
graph. It is clear that one can choose an optimal path to control
the capacitor and achieve a maximum starting torque as shown
in Fig.
5.
In the real world, however, it is difficult to realize
an adjustable mechanical capacitor. Therefore, a switched
capacitor control is proposed in the paper. The relationship
can be approximately expressed as
and
(35)
where Xceffective s the effective reactance, Xcrunnings the
reactance
of
the running capacitor,
T
is the addition of the
shorting interval and the turn off interval of the electronic
switch, and d is the shorting duty cycle of the switch.
Let X , = l/jwc, and substitute it into (34), then we can
obtain
Y
d = -
T
1
1 - d
ef iec t ive
=
Crunning-
(36)
When the switch is open,
d
is equal to zero to Ceffectives equal
to Crunning. n the other hand, when the shorting interval of
the switch is increased, the Ceffectiveill become larger. From
(36),
it is easy to adjust the value of the effective capacitor by
changing the shorting interval of the switch.
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LIU: A MAXIMUM TORQUE CONTROL WITH A CONTROLLED CAPACITOR FOR A SINGLE-PHASE INDUCTION MOTOR
21
TABLE I
PARAMETERS
F THE SINGLE-PHASENDUCTION MOTOR
Rqs
Xlqs
xmq
.
ds
'Ids
'rnd
Nd/Nq
RA;
X'
Iqr
'idr
Jm
'starting
'running
0.5420
2.40
0 .730
2 .030
18.220
50.810
1.67
0.5330
1.4S60
0.7240
2.020
0.0146 Kg-m2
300pf
40pf
N t m
1
_
3
-
o zo 1 400
ma
M O
I W O 1200
I 4 0 0 iaw
ILIOO r /min
Fig. 4.
Verification of torque-speed curve.
Fig.
4
hows the comparison of the torque-speed curves fo r
transient and steady state, when using method 2 to control
the SPIM. The parameters of the SPIM are shown in Table
I. Th e instantaneous torque-speed curve, which contains large
pulsation, is obtained from computer simulation that solves
the differential
(1)-(
1
I .
On the other hand, the sm ooth torque-
speed curve is obtained from the steady-state equivalent circuit
as shown in Fig. 2(c). It is clear that the steady-state torque-
speed curve is very close to the average torque-speed curve
which is obtained from the transient state. This can explain
why we can use the steady-state model to replace the transient
model to derive the control algorithm of the optimal path of
the SPIM.
The optimal starting curve, which expresses the relationship
between the motor speed and the optimal effective capacitor is
shown in Fig. 5.According to the figure, when the sp eed of the
motor is increased, the effective capacitor must be decreased
slowly to obtain the maximum torque. However, the relation-
ship between optimal effective capacitor and motor speed is
nonlinear, and their relationship depends on the parameters of
the SPIM. Traditionally, two capacitors are used to control
the same SPIM. A 300-pF starting capacitor and a 40-pF
running capacitor is installed, and then these two capacitors
are switched at 70% rated speed by a centrifugal switch. The
traditional method is simple and easy to implement, however,
it needs a centrifugal switch and a large starting capacitor.
50
O
m
150
loo
0 dm i n
Fig. 5. Optimal starting curve
of
capacitor
as
a function of motor speed.
I
w a& a8 a87 am 8
(b)
Fig. 6.
across capacitor
V,.
Measured
auxiliary winding wavefo rms. (a) Current zis b) Voltage
Moreover, the effective capacitor, while the motor is starting,
only changes at two different values.
A
switched capacitor
can chang e the effective capacitor and it provides an effective
method of overcoming the difficulty associated with the need
for a variable capacitor. In Fig.
5,
the effective capacitor is
changed from large to small when the speed of the SPIM is
increased. Then, it
is
clear from (36), that the shorting period
duty cycle is higher when the motor is in the standstill mode,
and then gradually decreases as the motor starts. When the
speed of the SPIM closes to synchronou s speed, the shorting
period duty cycle will reduce to zero.
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22
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS,
VOL.
42,
NO. 1,
FEBRUARY 1995
A
8 I
I
I
I
a84
085
086
om OBB om osos-
(b)
Fig.
7.
Measured waveforms of main winding. (a) Current i & .
(b)
Voltage
LYs.
Iv EXPERIMENTAL
ND SI MUL AT E D
RESULTS
The block diagram of the proposed system is shown in
Fig. 1. The capacitor voltage
V,
is detected by the isolated
amplifiers and the currents z , z are sensed by the Hall
effect current sensors. The speed of the motor is sensed by a
tachometer which aligns with the rotor. Some analog-to-digital
converters are used for converting the analog signals to the
digital code, which can be easily read by a microprocessor.
A microprocessor-based digital controller is proposed to
execute the optimization and control algorithms. A digital
signal processor
(DSP) TMS320-C30
is used to implement
the system. The DSP is 32 bit with a 60-11s sing le instruc tion
execution time. It also offers floating-point operation
[111.
The
hardware circuit is very simple because the optimization and
control algorithms are executed by the DSP. The motor is
a single-phase motor, the parameters of which are shown as
Table I. The solid-state switch in parallel with capacitor is
realized by IGBT's.
The hardware circuit consists of three parts: a zero voltage
detecting circuit, a sawtooth-wage generator and comparator
circuit, and an interfacing circuit. The sawtooth generator is
synchronized to the zero voltage detecting circuit. When the
voltage across the capacitor is detected to be zero, the sawtooth
generator begins to generate a linear voltage proportional to
time. This voltage is compared to a dc voltage, which comes
from the
DSP
and is proportional to the shorting period
command of the electronic switch. The electronic switch is
ms
6
Fig.
8.
Method 2.
Measured shorting period -, to speed curves. (a) Method 1. (b)
short-circuited until the sawtooth generator voltage is equal to
or greater than the dc voltage. When the electronic switch
is opened, the sawtooth generator keeps zero voltage until
the next zero voltage detecting signal is detected. Some
typical experimental results, which were obtained at
zlqs
=
154s in377t , y
= 3
ms, Tl =
0.1
N-m, are shown in Figs. 6
and 7. The current wave form of auxiliary winding,
zzs
is
shown in Fig. 6(a), and the voltage waveform of the switched
capacitor, V,, is shown in Fig. 6(b). It can be seen that
V,
lags
z
90 . The reason is that
i
can be considered as the current
passing through the effective capacitor. The main winding
waveforms of current and voltage are shown in Fig. 7(a) and
(b). Both of them are very close to sinusoidal waveforms.
The shorting period-speed curves of method 1 and method
2
are shown in Fig. 8(a) and (b). Although they have different
shorting period-speed curves, the shorting period
y
decrease to
zero when the speed is close to the rated speed. This is because
the value of the optimal capacitor in both curves is decreased
as the speed is increased. The torque-capacitor curves at
different fixed speeds and torque-speed curves for different
conditions and different methods are shown in Fig. 9.They are
obtained from computer simulation. According to Fig. 9(a),
the torque-capacitor is an arc when the speed is fixed. So,
there exists just one capacitor which can produce a m aximum
torque at any fixed speed. From Fig. 9(a), the optimal capacitor
must be changed at different speeds. The average torque-speed
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.-
23
IU: A MAXIMUM TORQUE CONTROL WITH A CONTROLLED CAPACITOR FOR A SINGLE-PHASE INDUCTION MOTOR
2 0 3 3 I I I I I I I
lam
1603
lha)
-
-
I I 1 I I
Nt-
3
t
'lmn
m . , , , , , , ( 1 1
1830 -
1m
m
1m
-
lox -
BM -
-
-
0 02 Ob 06 Q8 1.0
1.2
1L 16 U
20
2 2
(a)
rimin
sec
O t
D O 0
I
U
-
-
..
-
.--_
-...
-
..
i
1
-*o m ya 600 aa
1
1x10 IUIO 1600
laa
r l m i n
b)
Fig. 9. Simulated torque curves. (a) Torque-capacitor. (b) Torque-speed.
curves are shown in Fig. 9(b). Both method
1
and method
2 have
a
larger torque than that obtained by just adjusting
y at fixed periods.
For
the purposes
of
comparison, Fig. 10
shows the different methods for starting the same induction
motor. It can be seen that the use of a 3-ms fixed shorting
period to switch a running capacitor produces a very small
accelerating torque. However, when the capacitor is switched
in accordance with the principle proposed in the paper, the
acceleration time becomes shorter. The rise time of the SPIM
is
162
ms using method
1,
164 ms using method
2,
and
215
ms using capacitor runing and capacitor starting. The harmonic
currents
of
the auxiliary winding are produced by the switched
capacitor; however, the starting time required by the switched
capacitor is shorter than the traditional method.
Method 1 and method
2
can be realized by designing a
digital circuit to control the shorting period of
a
solid-state
switch, which is adjusted by the motor speed as shown in
Fig. 8.Then, the control method can become very simple and
then the
DSP
will be replaced by the digital circuit. Before
designing the digital circuit, the shorting period
y
related
motor speed w must be determined by using the proposed
1800-
1600
1400
1200
1000.
t 1 . n . -
50 l a ,
150 200
a0
MO 350 400 450 500
5 5 0ms
(b)
system in the paper. running capacitor with a solid-state switch. Since the switched
capacitor is dynamically varied, the optimum capacitor value
can be on-line adjusted to achieve the maximum torque at
different speeds. Two algorithms to obtain the maximum
torque of the SPIM are proposed here. Experimental results
validated the analytical results. Since the starting switch is
one
of
the most failure prone components of a single-phase
V.
CONCLUDINGEMARKS
The design and implementation of a switched capacitor
control for a single-phase induction motor has been presented
here. The starting capacitor and centrifugal switch can be elim-
inated by periodically and synchronously short-circuiting the
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS. VOL. 42, NO. 1, FEBRUARY 1995
induction motor, elimination of this device is expected to
significantly improve the reliability
of
such machines. The
results in this paper represent a first step toward the design
of
some optimal algorithms employing switched capacitor
techniques.
ACKNOWLEDGMENT
The author wishes to thank Pi-Chieh Wang for his help in
experimentation.
REFERENCES
[I]
C. A. Karybakas, G eneralized torque performance
of
two-phase induc-
tion motors controlled by TRIAC-An appro ximate analysis, IEEE
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40-45, Feb. 1981.
[2] L. M.
C.
Mhango and G. K. Creighton, Novel two-phase converter-fed
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[3]
J. D. aw and T. A. Lipo, A single phase induction motor voltage
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1, no.
4, pp, 24tb247, Oct. 1986.
[4] D. Alexa, Static frequency converter for supplying an asynchronous
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134, pt.
B, no. 1,
pp. 57-60, Jan.
1987.
[5] E. R. Collins,
H. B.
Puttgen, and
W .
E. Sayle, Single-phase induction
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I988
IEEE Cont Ind. Applicar., Pittsburgh,
PA, Oct. 1988, pp. 246-252.
[6] E. R. Collins and R. E. Ashley, Operating characteristics of single-
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I991
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MI,
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1991.
[7] P.
C.
Krause, Analysis of Elecrric Machinery.
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[8] T. A. Lettenmaier, D. W. Novotny, and T. A. Lipo, Single-phase
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1988.
Tian-Hua Liu (SSM89) was bom in Tao Yuan,
Taiwan, Republic of China, on Novem ber 26, 1953.
He received the B.S., M.S., and Ph.D. degrees in
electrical engineering from the National Taiwan
Institute of Technology, Taipei, Taiwan, in 1980,
1982, and 1989, respectively.
From 1984 to July 1989, he was an Instructor in
the Department of Electrical Engineering, National
Taiwan Institute of Technology. He was a Visiting
Scholar in the Department of Electrical and Com-
puter Engineering at the University of Wisconsin,
Madison, from 1990 to 1991. Since August 1989, he has been an Associate
Professor in the Department
of
Electrical Engineering, National Taiwan
Institute of Technology. His research interests include motor controls, power
electronics, and microprocessor-based control systems.