Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
MODELING AND SIMULATION OF SINGLE PHASE
DOUBLE CAPACITORS INDUCTION MOTOR
Sameer H. Khader
Palestine Polytechnic University, P.O.Box 198,
Hebron, Palestine.
e-mail ; sameer@ppu.edu
Abstract:
The aim of this paper is to study the performances of Single-phase Induction motor at
varying Pulse Width Modulation signals (PWM) and adjustable switching capacitance.
The control system is implemented in such a way that it allowed the change in motor speed
and loading level.
The attractiveness of this configuration is the elimination of pulse generating unit, and the
centrifugal switch. The pulses are generated based on the back e.m.f , and the duty period
of the electronic switches is controlled in such a way with purpose to obtain optimized
capacitor values in turn leads to maximum torque and efficiency. The electromagnetic
processes are modeled and the simulation results are presented and analyzed in order
to obtain optimized solution combining maximum torque with low capacitors values.
Matlab /simulink approach is implemented in processing the behaviors of Single
Phase Induction Motor, where the most of the machine parameters are accessible for
control verification purposes . Prototype model for result verification should be described
in future work.
Keywords: Simulation, Synchronous Motors, Induction Motor, Capacitor Motors,
Vector Computer Control, and PWM.
I. Introduction:
Single-phase induction motors (SPIM) are usually
low power machines and are widely used in industrial
and domestic appliances. This motor cannot run directly
from the mains because of it's electromagnetic
behaviors due to yield forward and backward
electromagnetic fields that lead to zero starting torque.
Usually SPIM is operated with auxiliary windings
and starting capacitors and sometimes with another
additional capacitor called run capacitor in order to
improve its performances [1,2].
In the SPIM, the most important problem is to
determine the appropriate capacitor values at any
loading level.
The capacitor values can be changed by switching
of parallel connected capacitors, either by switching of
electronic switches connected across the capacitor [3,4]
in the auxiliary winding. In order to increase the control
range capability a similar electronic circuit is connect in
the main winding as well shown on figure 1.
Therefore, any capacitors value can be obtained by
controlling the ON-OFF time of switches Q1, Q2, Q3,
& Q4 as well discussed in [5].
At the same time the development of electronic
power devices lead to working on so called static
converters presenting high power factor and low
harmonic distortion observed from the primary AC
source [6,7,8]. In this work the influence of motor
parameters and capacitor values are discussed in order
to determine the optimized operation at any loading.
Vector control of induction motor proved to be
practical and effective in SPIM drives [9], despite the
winding asymmetry in these motors, where extra
coupling between two stator windings and results in
unbalanced machine operation. This, in turn, produces
current and torque pulsations, additional looses and
limits SPIM drive applications.
Figure 1: Electrical Circuit of SPIM
ISSN: 1790-5125
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ISBN: 978-960-474-110-6
Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
Various solutions have been implemented
aiming at overcoming these drawbacks such as
Hysteresis current control, but this solution is
insufficient at light load condition which is the
case most of the time [10, 11].
Another method is called a current-double
sequence also proposed to eliminate the torque
pulsations, but it characterized by additional
complexity due to additional controllers and
extensive on-line computations.
on equivalent circuit of figure2-b with self and
mutual inductances as follows:
(1)
(2)
Where: VA, VB, RA, RB, LA, LB, and iA, iB are
main and auxiliary voltages, resistances, self
inductances, and currents respectively. LAR, LBR, ia,
ib are mutual inductance and rotor currents
respectively.
Equations (1) and (2) can be represented in
matrix form:
(3)
Where:
a)
Phase windings in stationary coordinate
systems
The rotor voltage equations can be expressed
in matrix form as follows:
(5)
The second rotor equation:
b)
Simplified equivalent circuit in q-axis.
Figure 2: Equivalent Circuit of SPIM.
(6)
II. Mathematical Modeling of SPIM in
General Frame:
The circuit inductances mentioned in eq.(5)
and eq.(6) can be represent in matrix form as
In order to simplify the system analysis, the
following assumptions and considerations were
made:
 The induction motor magnetic circuit is
considered to be linear;
 The net magnetic motion force MMF has a
sinusoidal distribution in the airgap with
neglected core losses and minimized harmonic
spectrum.
 The saturation effect of magnetic circuit is
negligible.
 The capacitor resistances are negligible.
follows:
Where Va, Vb, Ra, Rb, La, Lb, ia, ib are
voltages, resistances, self reactances of main and
auxiliary rotor windings respectively, and ωr is the
rotor angular velocity.
The Phase diagram of SPIM can be seen in
Figure 2, where the indexes "A" and "B" represent
the stator phases in d-q axis, and "a", "b" represent
the rotor phases, concentrating and displaced by π/2
electrical degrees. The voltage equations for both
stator and rotor circuits can be derived based
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The rotor equations (5), (6) and (7) can be
represented in matrix form as follows:
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ISBN: 978-960-474-110-6
Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
(13)
Where:
(8)
Where
.
.
Usually the rotor voltages mentioned in eq.(8)
can be assumed to Va=0, and Vb=0, and the
circuit inductances La, Lb are equal and represent
as Lr=La=Lb because of the squirrel cage
construction of the rotor.
.
=
The capacitor voltage of
the starting
capacitor can be expressed as follow:
;
=
=
;
(9)
;
Where Vcs, Rcs, and Cs are Capacitor voltage,
resistance, and capacitance respectively. Vs is the
supply voltage.
;
The motor electromagnetic torque is produced
by the interaction of stator and rotor fluxes, yields
by the circuit currents. The rotor movement
equation illustrates how the electromagnetic torque
varies as the motor loading changes, as follows:
(14)
, represent the number of stator
Where
and rotor windings respectively.
(10)
III. Mathematical Modeling of SPIM in
DQ Coordinate System.
Where Tem, TL, J, B are the electromagnetic
torque, loading torque, moment of inertia and
viscous friction coefficient respectively.
The electromagnetic torque is expressed as
follows:
The vector control of SPIMs needs a machine
model depicted in eq.(12) and (13) to be presented
in rotating reference frame entitled " DQ
Coordinate System" [12,13] as follows:
(11)
With purpose to eliminate the circuit asymmetry
due to differences in both auxiliary and main
windings, the motor parameters are referred to the
auxiliary winding.
The obtained mathematical model realizing such
transformation in matrix form is as follow [ 11]:
(15)
Stator parameters of eq.(3) :
(16)
(12)
Where:
Rotor parameters of eq.(8) :
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ISBN: 978-960-474-110-6
Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
.
torques1
torques
Torque pulse
-Ksig av e
rpm
average
Aux
voltage
.
vs
1
Te
Out1
Out2
wk
Power
1/J
CAPACITOR
MOTOR
Bm
is
wmec
speed
-K-
Te
In1
(17)
2
1
s
-K-
Demux
peak2rms
The d-q material resistances of eq.(15) and
eq.(16) are time varying resistances as well
mentioned in research paper [6] because of
unequal values of stator auxiliary and main
resistances. These terms can be presented as
follows:
magnitude
capacitor
voltage
-K-
signal
angle
current
current1
Fourier
-K-
v aux
ias
v cap
wmec
rpm5
aux wdg
currents
Figure 3: Equivalent Circuits Electrical Circuit of SPIM in
d-q references
The simulink circuit used in studying the motor
behaviors is illustrated on figure3, where the main
simulation blocks are based on designed package by
Riaz [14], and Matlab –Simulink package [15].
(18)
The produced electromagnetic torque in d-q
reference frame taking into account eq.(11), eq.(15)
and eq.(16) can be expressed as :
Several performances are discussed and
illustrated at various capacitor values and supply
voltage as follows:
(19)
Prod. Torque of SPIM
40
To simplify the torque equation, the mutual
inductance LAR, and LBR can be represent with
leakage and mutual inductance LAR=LlAR+Lma
LBR=LlBR+Lmb, and LARLBR;
therefore the
electromagnetic torque can be expressed as follows:
Torque, N.m
Cst=10uF
30
20
10
0
(20)
0
0.2
0.4
0.6
Where Lm= Lma= Lmb.
0.8
1
1.2
1.4
1.6
1.8
2
Torque & Speed of SPIM
1500
Actual Speed, rpm
IV. Simulation Results
The motor performance is simulated under two
modes of operation by using Matlab-Simulink. In
the first mode , the capacitor is held constant. In the
other mode a switching capacitor resulting in a
variable capacitance is discussed. This research is
done over a single–phase induction motor with the
following parameters:
Table. 1
VS
220 V
5.22 
9.54mH
4.02 
8.54mH
Lmb
180mH
Lma
180mH
B
0.001N.m/rad
f
50Hz
1400rpm
n
Tn
10N.m
2.12 
3.13mH
P
4
In
7.5A
0.75
J
0.0354kgm2
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1000
Cst=10uF
500
0
-500
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Figure 4: Electromechanical Performances at certain
Cst..
1. Motor speed and torque at various supply
voltage K=0,5..0,95, where K=VB/VS are displayed
on figure4. It's shown that significant effect of
varying the voltage is observed at starting of the
motor, while at steady operation, there is negligible
effect of varying this.
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Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
Actual Speed of SPIM
Produced & Reference Torque of SPIM
1400
35
Cst=10uF
Cst=60uF
Cst=110uF
Cst=160uF
30
S. off=0.50
S. off=0.75
1000
Actual Speed, rpm
Produced Torque, N.m
25
S. off=0.25
1200
20
15
10
S. off=0.90
800
600
400
200
5
0
0
-200
-5
0
0.2
0.4
0.6
0.8
1
1.2
Time, S
1.4
1.6
1.8
0
0.5
1
2
1.5
Time, S
2
2.5
3
2.5
3
a)Motor Speed
a)Torque
Run Wind. RMS Current
30
Torque VS Speed
35
Cst=10uF
Cst=60uF
Cst=110uF
cst=160uF
30
25
S.Off=25%
S.Off=50%
S.Off=75%
S.Off=90%
20
20
Irunr ms, A
Torque, Nm
25
15
10
5
15
10
0
5
-5
-200
0
200
400
600
800
1000
Actual Speed,rpm
1200
1400
1600
0
b)Mechanical Performance.
Figure 5: Motor Torque at different Cst.
0
0.5
1
1.5
Time, S
b)Run Winding rms current
Figure 6: Motor Speed , and current at different
values of switching ratio of auxiliary winding.
2. Motor speed and torque at different values of
starting capacitors Cst= 10, 60, 110, 160 uF, are
illustrated on figure5 a, b; where it is shown that as
the starting capacitance value increases, the starting
torque increases respectively .
Actual Speed of SPIM
1600
1400
Actual Speed, rpm
1200
3. The Motor speed and torque at different
switching-off ratio
of the centrifugal
switch . Figure 6 illustrates how the motor speed
changes as the switching-off ratio varies. Where it's
shown that as the
acts directely on the speed
stability of the motor, therefore the drawn run
winding current changes accordingly as well shown
on figure6 b. Better performance and speed stability
.
can be realized at large values of
Cst=10uF
Cst=60uF
Cst=110uF
Cst=160uF
1000
800
600
400
200
0
-200
0
0.5
1
1.5
Time, S
2
2.5
3
2.5
3
a)Motor Speed
Run Wind. RMS Current
30
25
Cst=10uF
Cst=60uF
Cst=110uF
Cst=160uF
Irunr ms, A
20
4. Motor speed and current at different values
of starting capacitor and
90%. Figure7
illustrates how the motor speed changes as the
starting capacitor varies. It's worth to mention that
small values of starting capacitor leads to high
drawn current, therefore it's necessary to connect an
appropriate capacitor value causing minimum
drawn current, fast speed building up and therefore
minimize the power consumption.
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2
15
10
5
0
0
0.5
1
1.5
Time, S
2
b)Run Winding rms current
Figure 7: Motor Speed , and current at different
values of Starting capacitor and 90% switching
ratio.
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ISBN: 978-960-474-110-6
Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
 The Motor speed and torque: Figure 9-a,b
illustrates such relations, where it's shown that the
motor speed reaches the rated limits at certain
starting capacitor value, and the developed torque
also being maximum at certain starting and run
capacitors values.
V. Capacitors Optimized Values
Because of some restrictions in selection of
capacitor values due to their high starting current,
size and cost, it's necessary to determine the
optimized starting and run capacitors values. The
following performances states how the motor
behaviors varies by changing the starting capacitor
at given run capacitor values, as well shown on
figure 8 and figure 9.
Motor Torque vs Starting Capacitor
18
Crun=40uF
Crun=80uF
Crun=120uF
Crun=160uF
16
 The drawn rms current and
speed at
different start and run capacitor values:
Figure 8-a,b illustrates such relation, where it's
shown that changing the run capacitor value affects
the drawn current, on the other hand there is an
optimized start capacitor value at which further
increasing in these values causing negligible change
in the current and speed. For present studying case
at given load, as Cst increases more than 80uF
there is no change in the current and kept at
minimum value.
Tem, N.m
14
12
10
8
6
20
40
60
80
100
120
Cst, uF
140
160
180
200
a)Torque
Output Power vs Starting Capacitor
1600
Total RMS Current vs Starting Capacitor
1400
30
Crun=40uF
Crun=80uF
Crun=120uF
Crun=160uF
25
1200
1000
Pout, W
800
Isrms, A
20
Cst=160uF
Cst=40uF
600
400
Crun=40uF
Crun=80uF
Crun=120uF
Crun=160uF
200
15
0
-200
10
-400
20
5
20
40
60
80
100
120
Cst, uF
140
160
180
40
60
80
100
120
Cst, uF
140
160
180
200
200
b)Output Power
a)Motor RMS Current
Figure 9: Motor output power and torque at
different values of start and run capacitor Speed ,
at 75% switching ratio .
Motor Speed vs Starting Capacitor
1500
1000
N, rpm
VI. Conclusion:
Simulation of SPIM with adjustable switched
double capacitor has been performed in transent and
steady state operation under variable loading
conditions. Matlab/simulink is used as simulation
tool, where the mathematicl model has been derived
and performed in Matlab environment. This desiged
model allows us to regulate the motor parameters
including capacitors values and oberving the
occurred electromgnetic behaviours. The obtained
results shows that the motor has optimized vlaue of
start and run capacitors that leads to maximum
output power with minimum power consumption.
In this model the centrifugal switch is replaced
by electronic converter that can be controlled by
DSP unit with purpose to obtian optimum capacitor
values resulting maximum torque and efficiency.
500
Crun=40uF
Crun=80uF
Crun=120uF
Crun=160uF
0
-500
20
40
60
80
100
120
Cst, uF
140
160
180
200
b)Motor Speed
Figure 8: Motor current and speed at different
values of start and run capacitor Speed , at 75%
switching ratio .
ISSN: 1790-5125
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ISBN: 978-960-474-110-6
Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
The next stage of this work is to build the
prototype model in order to verify the analytically
obtained results.
Machines, Oxford University Press, 1998.
[11]
R. Correa, M.B., Jacobina, , A.M., daSilva,
C.B., Lima E.R," Vector Control Strategies
for Single Phase Induction Motor Drive
Systems, IEEE Trans. On IA, Vol. 51, no.5,
pp.1073-1080, Oct. 2004.
[12]
Ozpineci B. And Tolbert L.M., Simulink
Implementation of Induction machine
Model, a Modular Approach, IEEE, 2003,
pp.728-734.
[13]
Novotony, D.W., Lipo, T.A., Vector
Control and Dynamics of AC drives",
Oxford Science Publications, 2003
Riaz, " Simulation Package for Studying
the performances of Induction Motors,
2006.
VII. References:
[1]
[2]
Sedat Sunter, Mehmet Ozdimer, Bilal
Gumus, " Modeling and Simulation of A
Single Phase Induction Motor with
Adjustable Switched Capacitor", 9th
International
Conference
on
Power
electronics and Motion Control- EPEPEMC, 2000, Poland.
B. Ozpineci, L. Tolbert, " Simulink
Implementation of Induction Machine
Model- A Modular Approach, IASIEEE,2003, Vol.1, pp.728-734.
[14]
[3]
Bimal K. Bose, Modern Power Electronics
and AC drives, Prentice Hall 2002.
[15]
[4]
Muljaldi, E., Zhao, Y., Liu,T.H., lipo,
T.A.," Adjustable AC Capacitor For a
Single induction Motor", proceeding IEEE
Conf. IA, Dearborn, Michigan, 1991,
pp.185-190.
[5]
Ozdemir, M., Sunter, S., Gumus, B.," The
Transient and Steady state performance of
Single-Phase Induction Motor with Two
Capacitors Fed by Matrix converter."
International Journal of Computation &
Mathematics in electrical Engineering,
Vol.17, Isue,2, 1998, pp.296-301.
[6]
Zadeh, Vaez, S., Harooni, Reicy, Sh.,"
Decoupling Vector of Single-Phase
Induction Motor drives", IEEE Trans on IA,
Vol.14, pp.733-738, 2005.
[7]
Chomat M., Lipo T.A., " Adjustable –Speed
Single-Phase IM Drive with reduced
number of Switches," IEEE Trans. On IA,
Vol.29, No.3, pp. 819-825, 1993.
[8]
R. Correa, M.B., Jacobina, C.B., Lima,
A.M. ," Field oriented Control of a singlePhase Induction Motor drive", Conf. rec. of
Annual IEEE Power Electronics Specialist
Conference, PESC 98, Vol.2, pp.990-996,
1998.
[9]
R. Correa, M.B., Jacobina, C.B., Lima,
A.M., da Silva, E.R," A Three-Leg Voltage
source Inverter for Two-phase AC Motor
Drive System", IEEE Trans. On IA, Vol. 17,
no.4, pp.517-523, Jul. 2002.
Vas. P, Sensorless Vector Control and
Direct Torque Control of Induction
[10]
ISSN: 1790-5125
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Matlab (7.01) and Simulink (6.5) Tutorial,
2004.
ISBN: 978-960-474-110-6