7/23/2019 Tian Hua1995

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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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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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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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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.

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[I]

C. A. Karybakas, G eneralized torque performance

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[3]

J. D. aw and T. A. Lipo, A single phase induction motor voltage

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[ IO ]

D. G. Luenberger,

Linear and N onlinear Programming.

Reading, MA:

Addison-Wesley, 1984, 2nd ed.

[ I l l

P. Papamichalis and R. Simar, Jr., The TMS320C30 floating-point

digital signal processor, IEEE Micro, vol.

8 ,

no. 6, pp. 13-29,

Dec.

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.