Various Characteristics of Induction Motor
by Harmonic Equivalent Circuit with LC Resonance
Keiju Matsui, Masanori Kondo , Masaru Hasegawa, Isamu Yamamoto
Chubu University, Matsumoto-cho, Kasugai-city,
Aichi, 487-8501, JAPAN
Abstract
This paper proposes a new equivalent circuit for induction motors having the harmonic components. The proposed circuit consists
of T-type equivalent circuit and LC resonant circuit, which is connected in parallel with load resistance. Various features of the
induction motors including harmonics can be easily obtained by the proposed circuit. The fundamental component is expressed by
the conventional output resistance, and the ripple elements of the mechanical output are represented by the current flowing in the
LC parallel resonant circuit.In the proposed equivalent circuit, the current flowing into the load resistance becomes sinusoidal, so
that we can separate ripple components from fundamental one. It is necessary to decide the appropriate values of L and C as the
resonant frequency that gives the fundamental frequency. Strictly speaking, the resonant frequency is denoted by rotor speed. By
means of this circuit, various harmonic torque characteristics can be easily obtained.
Key words: LC resonant circuit, induction motor, torque ripple, harmonic equivalent circuit
1 INTRODUCTION
Induction motors are superior to some other type motors in
terms of cost and robustness. The T-type equivalent circuit
of the induction motors is very useful as a circuit model
which can express various electric quantities such as
voltage, current, magnetic flux, torque and so on. As a
result, this circuit is generally used to calculate those
characteristics easily.
It is possible to obtain a mechanical output for single
frequency by using a general T-type equivalent circuit if the
waveform is sinusoidal. In a case that the waveform of the
input is distorted, however, the mechanical output can not
be expressed by T-type equivalent circuit. In this case of
distorted current, the output torque cannot be obtained by
calculation in T-type equivalent circuit, because the value of
load resistance R = (1-s)R2/s which represents mechanical
output varied with harmonic components. In such a way, the
harmonic makes the slip s varied according to its frequency
component. From such reasons, when mechanical output is
calculated by using T-type equivalent circuit for distorted
waveform, it is necessary to calculate for each frequency
component using equivalent circuit. In other word, it is
difficult to represent the T-type equivalent circuit in terms
of both the fundamental component and the ripple
components, simultaneously.
Under such background, this paper proposes a new T-type
equivalent circuit for induction motors where the harmonic
components can be calculated in a simple way. The
proposed circuit consists of T-type equivalent circuit and
LC resonant circuit which is connected in parallel with load
resistance R. Various features of the induction motors
including harmonics can be easily obtained by the proposed
circuit. The fundamental component is expressed by the
conventional resistance R, and the ripple elements of the
mechanical output are represented by the LC parallel
resonant circuit. In the proposed equivalent circuit, the
current flowing into the load resistance R becomes
sinusoidal, so that we can separate ripple components from
fundamental one. At the beginning of this study, we thought
that our harmonic equivalent circuit was entirely an original
one. However, we found that Ohnishi had proposed this
harmonic equivalent circuit [5]. Distinction is that ours is
taking into consideration of rotation of rotor. Consequently,
concept of LC resonance is different a little.
2 CONVENTIONALHARMONIC
EQUIVALENT CIRCUITS
Analysis of the performance of the induction motor can
proceed as if there were a series of independent generators
all connected in series supplying the motor. Each generator
would represent one of the voltage terms. The conventional
induction motor equivalent circuit is very useful in
calculating the performance of a motor under steady-state
operating conditions. Since each harmonic current will be
independent of all of the others, a series of independent
equivalent circuits can be used to calculate the complete
steady-state performance of an induction motor with
nonsinusoidal voltages applied.
i1
R1
X1
i2
E1
V1
R2
represent voltage, current, magnetic flux and torque. By
means of this model, various characteristic can be
calculated and resolved. Consequently, the model is widely
used since it is able to represent a physical phenomenon
sufficiently. In a case of resolving like the mechanical
output, if it is for a case of sinusoidal input waveform, we
can calculate in single frequency. In a case of distorted
waveform, however, since those contain various harmonic,
each slip is different between frequencies, the output
resistance representing the mechanical output is varied due
to the frequency. We should calculate in every frequency
components to resolve the corresponding characteristic. On
the background of this problem, in this paper, a novel
equivalent circuit for induction motor is proposed in Fig. 2.
In the figure, R represents the conventional mechanical
output, which is connected in parallel to an LC resonant
circuit that represents various harmonic behaviors. In such a
way, the figure represents a mechanical output by means of
the load resistance and the LC resonant circuit.
X2
⎛ 1 − s1 ⎞
⎜⎜
⎟⎟ R 2
⎝ s1 ⎠
XM
(a) Fundamental-frequency equivalent circuit.
ik
R1k
kX1
i2
kX2
R2k
k
Vk
⎛ 1 − sk
⎜⎜
⎝ sk
kXm
⎞
⎟⎟ R 2k
⎠
(b) Equivalent circuit for the kth time harmonic
4 SIMULATION RESULTS
Fig.1. Conventional induction motor equivalent circuit diagrams.
The conventional induction motor equivalent circuit is
shown in Fig.1[1]. X1 and X2 are the stator and rotor leakage
reactances at the supply frequency, and Xm is the
corresponding magnetizing reactance. The rotor slip with
respect to the fundamental rotating field, which is usually
denoted by s, is denoted by s1, to distinguish it clearly from
the harmonic slip in the figure. Hence,
s1 = (n1-n)/n1
(1)
where n1 is the synchronous speed of the fundamental
rotating field, and n is the actual rotor speed. As can be seen,
the circuit becomes very complicated to obtain the
performances of induction motors with nonsinusoidal
voltages including harmonics.
By using the equivalent circuit in Fig.2, a simulation
procedure was excuted. Fig.3(a) shows various waveforms,
without load for driven by the six-step inverter. Since the
load resistance is infinity that is no load, the fundamental
load current, iuR becomes zero. It can be seen that the input
current iu1 appears like the familiar current waveform for
driven by the six-step inverter. Fig.3(b) shows for loading
where slip is s = 0.1. Mechanical output can be represented
by calculation by eu3*iuR. This harmonic equivalent circuit
can resolve the usual characteristics of the induction motor
including harmonics in the power supply.
It is important to design suitably the LC resonant circuit.
Due to the sharpness of resonance of tank circuit, output
waveform varies, where Q is a quality factor of waveform
defined by Q = 1 R C L .
If Q is not large sufficiently, the output voltage and its
current waveforms are distorted from sinusoidal waveform.
Consequently, some error for various characteristic will
appear.
(
3 PROPOSAL OF NOVEL HARMONIC
EQUIVALENT CIRCUIT
The conventional T-type equivalent circuit for induction
motor is simple and excellent circuit model, which can
iu1
R1
L1
L2
R2
iu2
)
iuM
iuRP
iuR
e u1
e u3
LM
L
C
Fig. 2. Equivalent circuit for induction motor.
R
25
iu1[A] 0
-25
25
iuM[A] 0
-25
25
iu2[A] 0
-25
25
iuR[A] 0
-25
25
iuRP[A] 0
-25
300
eu3[V] 0
-300
5
∆τ[Nm] 0
-5
300
eu1[V] 0
-300
25
iu1[A] 0
25
iuM[A] 0
-25
25
iu2[A] 0
-25
25
iuR[A] 0
-25
25
0
-25
300
eu3[V] 0
-300
5
∆τ[Nm] 0
-5
300
eu1[V] 0
-300
(a) without load.
(b) for slip s=0.1.
Fig. 3. Various voltage and current waveforms.
4.1 PWM Control for Eliminating the Specified Harmonics
In order to reduce the torque ripple in the six-step inverter,
the inverter waveform is modified so as to have some
notches as shown in Fig.4 accompanied by those of
previously mentioned six step inverter to confirm the
predominance of this notched type inverter. In the figure of
the pole voltage, eup, when the voltage polarity is switched
over, a notch with α degree is given to the waveform.
Torque ripple, ∆τett in effective value is shown in Fig.5, and
investigated when the notch α is varied, where the
fundamental component of output phase voltage is kept to
300
eup[V] 0
-300
25
iu1[A] 0
-25
25
iuRP[A] 0
-25
300
eu1[V] 0
-300
300
eu-v[V] 0
-300
5
∆τ[Nm] 0
-5
α
α
(a) α=16.3 degrees.
be 163.3V, that is, the line effective voltage is 200V. The
induction motor is driven with no-load. The figure shows
the torque ripple characteristic for various waveforms. As
notch width α is widening from zero, the effective torque
ripple is gradually decreasing. When α becomes 16.3 degree,
it takes the minimum value of torque ripple that is the well
known result. Beyond the minimum value, ∆τett is
increasing again, because the waveform of output voltage is
distorted significantly. An excellent analysis method was
already proposed [2], and those results are agreed well with
that in our paper’s method.
300
eup[V] 0
-300
25
iu1[A] 0
-25
25
iuRP[A] 0
-25
300
eu1[V] 0
-300
300
eu-v[V] 0
-300
5
∆τ[Nm] 0
-5
(b) six-step inverter drive.
Fig. 4. Various waveforms of PWM control.
α β
∆τett [Nm]
1.2
for six-step inverter
eup
1.1
Fig. 6. PWM voltage waveform for eliminating
fifth and seventh harmonics.
1.0
by Murai
16.3
0.9
25
20
15
10
5
α degree [°]
Fig. 5. Torque ripple characteristic with notches.
4.2 Torque Ripple Driven by PWM Inverter Eliminating
Fifth and Seventh Harmonic
According to reference [4], let us discuss about a strategy
eliminating the fifth and the seventh harmonics. In order to
confirm a validity of the method, the proposed one is
compared with that in such article. The pole voltage of the
six-step inverter is switched over as shown in Fig.6,
representing the pulse widths of notch by α and β degrees.
By means of such values, the torque ripple was calculated
by simulation, whose waveform is shown in lower side in
Fig.7, while the upper side in that is by Murai. In this center,
however, the resonant frequency is fr=48Hz which is for the
proposed one, but is f1 is a little below the supply frequency.
This torque ripple is agree well compared to the
conventional one by Murai. In the proposed method,
however, the magnetizing current is assumed to be ideal
sinusoidal one, so there is some error among them. Actually,
the magnetizing current is a little distorted from an ideal
sinusoidal one. In the simulation, the circuit constants are
R1=1.15, L1=1.6mH, LM=60.8mH, R2=1.08, and, L2=1.6mH.
The widths of notches are α=16.25 degree and β=5.82
degree. The induction motor is driven with 4% load
operation. From the appearance of the waveforms, we can
say that a satisfactory result can be obtained.
0
0.5
by proposed 0
fr=48Hz
0.5
by proposed 0
0.5
fr=50Hz
0
π
2π
Fig. 7. Comparison of ∆τ between
the conventional and proposed method.
4.3 Comparison Between Six-Step Inverter
and PWM Inverter Drives
Fig.8 shows various waveforms for PWM inverter, an
accompanied by the six-step inverter drive for comparison.
Carrier frequency for PWM is fc, 1980Hz, where the
fundamental amplitude of the output phase voltage is
adjusted so as to be 163.3V, that is, the line voltage is 200V
in effective value. The motor is driven with no-load. Fig.9
shows torque ripple characteristic as the modulation factor
is varied from α=0.5 to α=2.0. The PWM inverter is
controlled by sinusoidal modulation. At α=1.0, the carrier
triangular wave and the sinusoidal modulation wave
coincide with each other. Quality factor of waveform for the
usual PWM waveforms has the optimum and minimum
distortion factor about at α=1.0. In the torque ripple
characteristic, however, an optimum or minimum ripple
point is shifted a little and obtained at α=1.6. It is an
interesting result. The reason is that even at the
over-modulation region that is fairy large than 1.0, the
increased quantities of harmonic is not so much, but the
fundamental amplitude is significantly increasing relative to
that of the harmonic component.
300
eu-v[V] 0
300
eu-v[V] 0
-300
300
-300
300
eu1[V] 0
eu1[V] 0
-300
25
-300
25
iu1[A] 0
iu1[A] 0
-25
5
-25
5
∆τ[Nm] 0
∆τ[Nm] 0
-5
-5
(a)
(b)
Fig. 8. Various waveforms for PWM inverter (a), for six-step inverter (b).
but the envelope of the ripple is much pulsated every one-sixth
in a period. In Fig.10, the simulation constants are, α=0.3,
Ed=256.5V, carrier frequency, fc=1980Hz, fundamental
frequency, f1=20Hz. In Fig.11, α=0.9, Ed=85.5V, fc=1980Hz,
f1=20Hz.
1.8
∆τett [Nm]
1.4
for six-step inverter
1.0
300
eu1[V] 0
0.6
-300
25
iu1[A] 0
0.2
0.5
1.0
1.5 1.6
-25
25
2.0
modulation factor α
Fig. 9. Torque ripple vs. modulation factor.
In such a way, Fig.8 (a) shows this optimum waveforms at
α=1.6. It can be seen that the torque ripple is much reduced
and the output PWM waveforms, eu1 appear uniformly
modulated in the phase voltage waveform even at the
over-modulation region.
iu2[A] 0
-25
5
∆τ[A] 0
-5
Fig. 11. Various waveforms PAM+PWM strategy.
4.4 Torque Ripple Characteristics Driven at Constant V/f
In order to estimate their qualities on output waveform and on
the torque ripple due to PWM modulation strategies, the relative
characteristics were compared. With keeping the dc link voltage
constant, the modulation factor is adjusted as the supply
frequency is varied, that is termed “Modulation A” in this section.
With keeping the modulation factor constant, the dc link voltage
is adjusted as the supply frequency is varied, that is named
“Modulation B”. It can be seen that according to the difference
of modulation strategies, the appearance of torque ripple is also
different. In the torque ripple waveforms, it was anticipated in
advance that the amplitude of torque ripple in Modulation A is
larger than in Modulation B, and the actual obtained result
appears in such a way. In Modulation B, on the other hand, the
amplitudes of the torque ripple over the whole period is reduced,
300
eu1[V] 0
4.5 Starting Current Responses
In this LC resonant circuit, we are now considering to introduce
the mechanical operations. By representing the rotating speed in
the resonant frequency, the actual transient response could be
described to a certain extent. As the LC resonant circuit is
assumed to represent the rotational speed, the resonant frequency
should be varied according to the rotation. According to the
measured rotation speed, the resonant frequency was varied. In
Fig.12 shows inrush current when the whole voltage is applied at
the stationary. It can be seen that the experimental result and
simulated ones are agree well each other.
0
100ms
-300
25
(a) for experiment waveforms.
iu1[A] 0
-25
25
iu2[A] 0
-25
5
∆τ[A] 0
-5
Fig. 10. Various waveforms in natural sampling PWM.
0
(b) proposal method.
Fig.12.Starting current responses of induction motor.
5 CONCLUSIONS
In this paper, a novel T type equivalent circuit for induction
motor driven by the inverter with various harmonics. The
conventional T type equivalent circuit is accompanied by a LC
resonant circuit. The validity of this circuit is confirmed by
simulation, by means of comparison with the usual analysis
method.
The proposed equivalent circuit model for inverter driven
induction motor is effective for the steady state condition, not for
dynamic response and the like, because when the external circuit
supplies some electric power into the LC resonant circuit the
circuit can not store the mechanical energy. The idea is originated
from a short circuit theory of harmonic operations for induction
motor drive.
ACKNOWLEDGEMENT
This research was partly supported by a grant of the Academic
Frontier Promotion Project from Ministry of Education, Culture,
Sports, Science and Technology.
REFERENCES
[1] J.M.D. Murphy and F.G. Turnbull, “Power Electronic
Control of AC Motors”, Pergamon Press, p.225-228, 1988.
[2] Klingshirn, E.A., and Jordan, H.E., “Polyphase induction
motor performance and losses on nonsinusoidal voltage
sources”, IEEE Trans. Power Appar. Syst., PAS-87, 3,
Mar. 1968, pp. 624-631.
[3] Robertson, S.D.T., and Hebbar, K.M., “Torque pulsations
in induction motors with inverter drives, IEEE Trans. Ind.
Gen. Appl., IGA-7, 2, Mar./Apr. 1971, pp. 318-323.
[4] Y. Murai, S. Sugimoto, Y. Tunehiro, K. Iwasaki, “Simple,
analytical method of waveform of torque of induction
motor driven by PWM inverter”, IEEJ, p.661, 1981.
[5] Tokuo Ohnishi, “Modified Equivalent Circuit of
Induction Motor Driven by Periodic Wave Power
Source”, TIEEJ, vol. 110-D, no.3, p.301, 1990.
Masanori Kondo was born in Aichi
Prefecture, Japan, on July 27, 1979. He
received the B.Eng. in electrical
engineering from Chubu University,
Kasugai, Japan in 2002, where he is
currently a student of Graduate Course
in Electrical Eng., Chubu University,
and dose research on static power
converters. He is a student member of
the Institute of Electrical Engineers of Japan.
Keiju Matsui was born in Ehime
Prefecture, Japan, on September 20,
1942. He received the B.Eng degree
in electrical engineering from
Ehime University, Matsuyama,
Japan, in 1965, and the Dr.Eng.
degree from the Tokyo Institute of
Technology, Tokyo, in 1982. Since
1965, he has been with the
Department of Electrical Engineering, Chubu University,
Kasugai, Japan, where he is currently a Professor and is
engaged in research on static power converter. Dr.Matsui
received the Price Paper Award from the Institute of
Electrical Installation Engineers of Japan in 1997 and the
Outstanding Book Award from the Institute of Electrical
Engineers of Japan in 1999. He is a member of the Institute
of Electrical Engineers of Japan and the Society of Institute
and Control Engineers of Japan.
Masaru Hasegawa was born in Gifu,
Japan, in 1972. He received the B.Eng.,
M.Eng., and D.Eng., degrees in
electrical engineering from Nagoya
University, Japan, in 1996, 1998, and
2001, respectively. He is currently a
Lecturer in the Department of
Electrical
Engineering,
Chubu
University, Japan. His research
interests are in the areas of control
theory and application to AC motor drives. Dr.Hasegawa
received some awards from several foundations in Japan.
He is a member of the Institute of Electrical Engineers and
Society of Instrument and Control Engineers of Japan.
Isamu Yamamoto was born in Mie
Prefecture, Japan, in 1968. He received
the B.Eng., M.Eng., and Dr.Eng.
degrees from Chubu University,
Kasugai, Japan, in 1991,1993, and
2002 respectively. In 2002, he joined
as Researcher in the Academic Frontier
Promotion Project, Chubu University.
His research interests are power
electronics in a power system. Dr.Yamamoto is a member of
the Institute of Electrical Engineers of Japan.