Single-Phase and Asymmetrical Three-Phase Induction Motors:
A Comparative Steady-State Analysis Under Single-Phase Feeding.
L. Martins Neto,Dr.
J.R. Camacho,PhD
D.A. Andrade, PhD R.G. Mendonça, MSc.
Universidade Federal de Uberlândia
Electrical Engineering Department
P.O. Box: 593
38400.902 - Uberlândia - MG - Brazil
E-mail: jrcamacho@ufu.br
Abstract
In Brazilian rural areas single-phase power supply is very
common. Single-phase induction motors with reasonable
power are used to run farming activities which often,
depending of their size, is not an inexpensive option. The
aim of this work is to provide to farmers a simple,
inexpensive and reliable option by running a modified threephase induction motor [3] with as little structural
modification as possible in order to run it with the available
single-phase power supply. The asymmetrical three-phase
induction machine fed by a single phase source, already
discussed in previous work [1], has nothing more than an
ordinary squirrel cage rotor and a stator where three-phase
windings are located. These windings have the ordinary
spatial design with a displacement of 1200 between them,
but with a different number of turns in each phase. With a
proper relation of number of turns in each phase and with
the help of a capacitor (Cap) connected between phases B
and C, as in Figure 1, can be possible to make the
asymmetrical induction motor to produce nominal power at
nominal speed.
1 Introduction
This paper will present the following structure: discussion
of the mathematical modeling, study for the steady state
conditions, and related aspects, presentation of results
through digital simulation and performance comparison
with a similar symmetrical induction motor. Digital
simulation takes in account time and space harmonics for
the symmetrical and asymmetrical induction motors [6].
Results reveals that on selected conditions it is possible to
get reasonable performance out of the motor without
changing its original structure, the slot number is an
example.
A three-phase asymmetrical induction motor with
single-phase feeding can be easily derived in a production
plant from ordinary three-phase induction motors. Estimated
production costs doesn’t change substantially in both cases.
The only differences for both motors are the number of
windings in each stator phase; they will be different from
each other, and the mandatory presence of a capacitor as
shown in Figure 1.
Taking into consideration that in Brazil the single-phase
induction motor come out of a production line much more
expensive, for the same power, than the ordinary threephase induction motor, and this price difference increases
with power. Initial considerations made for the
asymmetrical induction motor led us to think that the
cost for such a motor will be in between the costs for the
single-phase and three-phase induction motors,
considering again, for both, the same output power. We
can say that, without expensive computations, cost will
be very close to that of the three-phase induction motor.
We only need to know if technically the asymmetrical
induction motor has a similar or better performance
when compared with the single-phase machine.
Figure 1. General test scheme.
The aim of this work is to demonstrate with
theoretical and practical analysis that the asymmetric
three-phase induction motor, with single-phase feeding,
has a performance similar to the single-phase induction
motor. Thinking in the global aspect, and taking into
consideration the machine volume, weight and other
characteristics, its performance certainly can be superior
to the single-phase machine.
2 Symmetrical Induction Motor Operating Point at Rated Load
Through steady-state tests on the ordinary induction
motor was observed that, when comparing symmetrical
three-phase induction motor data at rated load, the
induction motor doesn't develop 2 HP under 1720 rpm
with 4 A phase current (plaque data). Under the above
nominal condition of 4 A phase current, the resulting
speed was 1734 rpm. The average value for the net
electromagnetic torque was 7.02 N.m, obtained with a
torque-meter. Under no-load condition the average
electromagnetic torque developed in this case was 7.52
N.m. Under those circumstances it was necessary to
define a nominal operating point. The main constraint
being the fact that we can't exceed the specified plaque
current under steady-state conditions. The condition,
where the motor develops phase current of 4 A, is
defined as steady-state operating point. This means that
the nominal load condition was defined as, "Phase current: 4
A, speed: 1734 rpm, power: 1.73 HP, voltage: ∆ - 220 V/ Y
- 380 V and torque: 7.52 N.m".
3 Asymmetrical Induction Motor
3.1 Sinusoidal Steady-State Basic Equations
Accordingly to references [3] and [4], the mmf
(magnetomotive force) three-phase system MMFa, MMFb
and MMFc, produced by stator currents Ia, Ib and Ic, can be
decomposed in symmetrical components of zero sequence
(MMFa0), positive (MMFa1) and negative (MMFa2) which
allow us to write:
& a
 MMF
 & 
 MMFb  =
& c 
 MMF
&
1 1

1   MMFa
0




2
&
α   MMFa1 
1 α
&
1 α α 2   MMFa

2
 Na

0
 0
0
Nb
0
& 
0   Ia
 & 
0   Ib 
& 
Nc  Ic
(2)
Considering
asymmetric
stator
winding
( N a ≠ N b ≠ N c ), the symmetrical MMF components are
produced by asymmetrical current components, for “a”
phase they have the following names: Iaz, Iap and Ian,
respectively zero, positive and negative sequence
asymmetrical components. Thus can be written that:
&
& 
 MMFa
1 0 0  Iaz
0
 &


 & 
 MMFa1  = Na 0 1 0  Iap 
&
& 
 MMFa

0 0 1  Ian
2

1
1

2
b bα
c cα
& 
1   Iaz
 & 
bα   Iap

2  & 
cα   Ian
(3)
& = Z& ′an. Ian
&
Ean
(7)
From Equations (4) and (7) we have:
&a
& p
& n   Ia
& 
V&a   Zs
Za
Za
 z 
&  &
2 &
&
&
Vb  =  Zsa b α Za p b αZa n b   Ia p 
& a c α Za
& p c α 2 Za
& n c  Ia
& 
V&c   Zs

 n
(8)
& a = Rsa + jXsa
Zs
& = Zs
& + Z& ′a
Za
p
a
p
&
&
&
Za n = Zsa + Z ′a n
And making:
Rsb = b2 Rsa; Rsc = c2 Rsa; Xsb = b2 Xsa; Xsc = c2 Xsa.
From Figure 1 can be written that:
V&b − V&c = − jX cap . I&cap
I&b = − I&cap
& − Vc
&
V& = Va
1
X cap =
2. π .60. Cap
(9)
From the set of Equations (8) and (9) can be
unbalancing factor
T.
F&
& , Ic
& ,
& , Ib
Ia
the
and the electromagnetic torque
&
& = V (1 − γ + F& (1 − γ ))
Ia
1
2
Z&
&
& = bV (α 2 − γ + F& (α − γ ))
Ib
1
2
&
Z
&
& = cV (α − γ 1 + F& (α − γ 2 ))
Ic
&
Z
(10)
(4)
Where: b = Na/Nb and c = Na/Nc".
Voltage electrical circuit equation can be written as:
V&i = (Rsi + jXsi ). I&i + E&i
(6)
obtained phase current phasors
From Equations (1), (2) and (3) can be obtained the
following expression:
& 
 Ia
& 
 Ib  =
&
 Ic

& = Z& ′ap. Iap
&
Eap
Where:
(1)
The dot above the parameter name indicates phasorial
form and á = 1 /120o.
& a
 MMF
 & 
 MMFb  =
& 
 MMF
c
positive sequence impedances relative to the
magnetization branch (equivalent circuit) and rotor
referred to the stator.
So we have:
(5)
Where Vi and Ei, Rsi and Xsi are voltage and emf
(electromotive force) phasors, resistance and leakage
reactance in phase i (i = a, b, c - stator phases) respectively.
Emf’s from stator phases can be decomposed in positive and
negative asymmetrical components, for phase “a” they can
be written as: Eap and Ean. Relationships between positive
and negative components give origin to the negative and
γ1 =
1 + bα 2 + cα
1 + bα + cα 2
; γ2 =
;
1+ b + c
1+ b + c
α 2 α 
 1 1 & ; &
& ;

Z& 0 =  −  Zs
Z
=
−  Za
 b c a 1  b
c p
α α 2 
& ;
 Za
Z& 2 =  −
c  n
b
&
Ia
Z& − Z& 0γ 1 + jXcap. b.(γ 1 − α 2 )
F& = & n = &1
Ia p
Z 0γ 2 − Z& 2 + jXcap.b.(α − γ 2 )
(11)
α  & ;
1  & ; &
Z =  − 1 Za
Z& 3 =  − 1 Zs
c  a 4  c
 p
α2

& p;
− 1 Za
Z& 5 = 
 c

Z& = Z& 4 − Z& 3 .γ 1 + ( Z&5 − Z& 3 .γ 2 ) F&
& ] − Re[ Za
& ]| F& | 2 
3V  Re[ Za
p
n


T=
& |2
WS 
Z
|


2
WS = (1 − s)WR
where WR is the rotor speed.
3.2 Asymmetrical Motor Design
The asymmetrical induction motor prototype design was
developed from an exhaustive number of tests and motor
equations including time and space harmonics.
Equivalent circuit parameters from the three-phase
symmetrical induction motor should be used as a
benchmark, and from the torque equation (frequency
domain) it is possible to obtain results for the
electromagnetic torque "T", under nominal speed of 1734
rpm, and with a selected number of values for "b", "c" and
"Cap".
The following figure shows parameterized curves of "T"
against the value of "Cap" for several values of "b" and "c".
stator slot area for coil placement in the stator slots.
Remembering that one of the advantageous aspects of
this motor is that changes won't be necessary in its
mechanical structure.
3.3 Obtaining design parameters
Basic elements to obtain step and distribution factors
were obtained from the original three-phase symmetrical
induction motor winding, this data should be applied
numerically to digital simulation. Those values are
considered the same for both motors (symmetrical and
asymmetrical), since the only difference between them is
the number of turns in two of the phases. This difference
is assumed as not making any difference in such factors.
From the available data (motor identification plaque)
for the stator winding can be obtained basic elements to
define step and distribution factors. Such obtained
values are: coil step = 180° electrical degrees, adjacent
slots angle = 20o electrical degrees and number of coils
per phase = 3.
With the analysis of obtained torque oscillations, the
chosen group of values for "b, c and Cap" to build the
asymmetrical induction motor prototype was: b = 1.67, c
= 0.71 and Cap = 40µF which corresponds to all figures
shown in this work.
Taking those values as a base the number of turns of
phases "b" and "c" relatively to phase "a" can be
obtained. These values, where b=Na/Nb and c=Na/Nc,
are: Nb = 0.60xNa and Nc = 1.40xNa.
Through frequency domain modeling, for such
design condition, currents in the three-phases can be
obtained, which are: Ia = 4.27 A, Ib = 4.63 A and Ic =
3.25 A.
At the original induction motor the stator winding
has 42 turns in each phase with 0.65 mm2 wire.
Considering the asymmetrical induction motor, wires are
changed and a prototype can be built from the
dimensions in Table 1.
Table 1
Phase
Conductor
Turns/phase
A
0.65 mm2
42
B
0.65 mm2
25
C
0.32 mm2
59
3.4 General Test Scheme
Figure 1 - Capacitance optimization. Parameterized curves
of "T" for b = 1.67 and c in the range from 0.51 to 1.21.
From those curves can be seen that for each
composition of "b" and "c" an adequate capacitor exists
under electromagnetic torque nominal condition of 7.52
N.m. It will possible, them, various possible designs, since
curves from c=0.51 to c=0.91 presents possible nominal
torque situations.
Therefore, in order to obtain relations "b", "c" and
"Cap" to build the prototype the decisive constraint will be
the torque behavior taking in consideration its steady-state
oscillation, under the cases where will be possible to modify
the phase coils. It is important to point out that values of "b"
and "c" will be constrained and can't be well above the
unity. The reason is that the space available will be limited
to the original symmetrical three-phase induction motor
While doing the tests the following measurement
equipment has been used as shown in Figure 1:
Data acquisition system (A-D converter) - which
receives analog voltage signals:
1 - from single-phase feeding: voltage (channel 1) and
current by means of a shunt (channel 2);
2 - from speed meter (channel 3),
3 - from torque meter (channel 4).
Shunt - 15 A - 60 mV.
Torque-meter - consists of a cylindrical steel ring
with a thousand of millimeter thickness, where in the
external surface is placed a "strain gage" bridge. Elastic
deformation of cylindrical ring is transferred to the
bridge by the motor toque action. The bridge feeding,
together with a voltage amplifier, is the origin of a small
voltage in the terminals of the torque-meter proportional
to the bridge resistance variation and for instance
proportional to the motor torque.
Electromagnetic brake - eddy currents braking system
with the following plaque data:
Model - AB703
Voltage - 90 Vdc
Current (hot) - 1.70 A
Ohm at 20° C - 40.5 Ω
Power - 7.5 CV
Speed - 1800 rpm
4 Theoretical and Experimental Results
The asymmetrical motor prototype was built from an
ordinary commercial three-phase induction motor. No
changes were made in the motor structural design when
changing from symmetrical to asymmetrical. Ordinary tests
were applied to the symmetrical motor, and parameters
obtained were suitable for application in previously shown
modeling equations. The asymmetrical motor should have
the same nominal speed and torque as the original
symmetrical motor. Those values were respectively 1733
rpm and 7 N.m.
After optimization studies for the many possible cases,
were found the best combinations for the stator winding
relationships and capacitor given previously. In those
conditions stator currents resulted in the nominal values
shown below. Where the remaining asymmetrical induction
motor nominal values are: T (N.m) = 7.0, W (rpm) = 1733,
V (Volts) = 380.
1. Asymmetrical induction motor designed and built
with the theoretical values from table 1 as a base.
2. Single-phase commercial induction motor with the
auxiliary winding disconnected with a centrifugal
switch after starting.
3. Single-phase commercial induction motor with
auxiliary winding and capacitor permanently
connected.
The three motors have the same nominal expected
output power. Results obtained are presented in Table 1.
Motor type
Feeding
T
W
Efficiency
V (V)
I (A)
PF
Nxm
Rpm
%
1
380
4.22
0.97
7.03
1733
83.5
2
220
10.0
0.74
7.28
1737
78.5
3
220
7.28
0.92
7.17
1765
88.0
Table 1 - Practical results for induction motors: (1) 3φ
asymmetric, (2) single-phase with starting capacitor and
(3) single-phase with permanent capacitor.
Figure 5 - Asymmetrical induction motor speed.
Figure 3. Simulated torque (zoomed) with the fundamental
only - b = 1.67, c = 0.71 e Cap = 40 ìF.
Figure 6 - Current for the asymmetrical motor at rated
load.
Figure 4. Experimental torque - b = 1.67, c = 0.71 e Cap =
40ìF.
Three distinct prototype motors were tested in the lab:
For Figure 5 we have Tm=7.52 N.m with ∆T=4.65%
e Wr=1733 rpm.
5 Conclusion
Biography
The main conclusion is due to the fact that the asymmetrical
motor can compete with the single-phase induction motors.
Comparing with a type 2 induction motor, the asymmetric
motor has great advantages concerning energy savings and
power factor control. Comparing with a type 3 induction
motor, considering the fact that this motor has two phases
internally, its characteristics are very close to the
asymmetrical motor. However, it has the disadvantage of a
bigger volume, and the advantage of better efficiency since
it works with speed a little bit higher. However, this small
increment in efficiency can be expensive to be obtained in
the case of a single-phase with a permanent capacitor
compared with the asymmetrical induction motor with
single-phase feeding.
The most important practical fact is that the
asymmetrical induction motor shown here is working in
farming activities for quite some time as a solution for
single-phase power supply.
Another important fact to be mentioned is that for this
solution it is not necessary any structural changes in the
three-phase ordinary induction machine, stator and rotor slot
numbers remains as in the original design. Structural
changes in the motor are expensive and require
modifications in the whole motor design.
The above observed facts suggest that in the case of
asymmetrical induction motor the little changing in design
characteristics was very much an optimization problem. It
will be also technically possible to build asymmetrical
induction motors larger than 10 HP.
Luciano Martins Neto - Dr. Martins Neto was born
in Botucatu, SP, Brazil in 22/05/48. He has a Doctoral
degree in Mechanical Engineering from Escola de
Engenharia de São Carlos at Universidade de São Paulo
(USP), São Carlos, Brazil since 1980. Worked as a
lecturer at Faculdade de Engenharia de Lins, Lins, SP,
Brazil, at Escola de Engenharia de São Carlos ( USP),
São Carlos, Brazil and at the Electrical Engineering
Department (UNESP - Universidade Estadual Paulista)
at Ilha Solteira, SP, Brazil. He is working as a Senior
Lecturer at Universidade Federal de Uberlândia, MG,
Brazil. His areas of interest are Electrical Machines and
Grounding.
José Roberto Camacho - Dr. Camacho was born in
Taquaritinga, SP, Brazil in 03/11/54. Completed his
PhD degree in the Electrical and Electronic Engineering
Department at Canterbury University, Christchurch,
New Zealand, in August 1993. He is a Senior Lecturer at
Universidade Federal de Uberlândia where he works
since February 1979. Dr. Camacho is a ResearcherConsultant of CNPq (Brazilian National Council for
Scientific and Technological Development) and
collaborator-member for the Brazilian Commitee of
CIGRÉ-JWG 11/14-09 (Unit Connection). His areas of
interest are Dynamic Simulation, Electrical Machines
and HVAC-DC conversion.
Darizon A. Andrade - Dr. Andrade was born in
Monte Alegre de Minas, MG, Brazil in 1956. He
completed his MSc in Electrical Engineering from UFU
- Universidade Federal de Uberlândia, Uberlândia, MG,
Brazil in 1987. He obtained his PhD in Electrical
Engineering at the University of Leeds, Leeds, UK in
1994. He is a Senior Lecturer at UFU where he works
since 1980. He currently is the Head of Graduation
Courses in the Electrical Engineering Dept. at UFU. His
areas of interest are Electromechanical Energy
Conversion, Non-Conventional Electrical Machines,
Vector Control and Drive Systems.
Roberlan G. de Mendonça - Mr. Mendonça was
born in Itabuna, Bahia, Brazil in 06/05/69. He finished
his BSc in Electrical Engineering from UNIVALE Universidade do Vale do Rio Doce, Governador
Valadares, MG, Brazil in 1991. Finished his Master’s
degree in 1997 at UFU. Currently, he is working
towards his Doctoral degree at Universidade Federal de
Uberlândia, his area of interest is Electrical Machines.
References
[1] Martins Neto, L; Teixeira, E.P. & da Silva, R.F.,
"Performance Control of Asymmetric Three-Phase
Induction Motors With Single-Phase Power Supply - A
Neural Network Approach", ICEM-94 - International
Conference on Electrical Machines, September 1994,
Paris, France.
[2] Tozune, A., "Balanced Operation of Three-Phase
Induction Motor with Asymmetrical Stator Windings
Connected to Single-Phase Supply System"; IEE
Proceedings-B, Vol. 138, n. 4, pp. 167-174, July 1991,
London, UK.
[3] Martins Neto, L., "Three-Phase Asymmetrical
Induction Motor"; The First International Conference
on Power Distribution, Belo Horizonte, MG, Brazil,
1990. In Portuguese.
[4] Tindall, C.E. & Monteith, W., "Balanced Operation
of Three-Phase Induction Motors Connected to SinglePhase Supplies"; Proceedings IEE, Vol. 123, n. 6, pp.
516-522, June 1976, London, UK.
[5] Straughten, A. & Tracy G.F., "Single-Phase
Operation of Three-Phase Induction Motors",
Engineering Journal, pp. 14-17, February 1969.
[6] Martins Neto, L.; Camacho, J.R.; Salerno, C.H. &
Alvarenga, B.P., "Analysis of a Three-Phase Induction
Machine Including Time and Space Harmonic Effects:
The A, B, C Reference Frame", PE-154-EC-0-10-1997,
recommended and approved for publication in the
IEEE Transactions on Energy Conversion, by the IEEE
Electric Machinery Committee of the IEEE Power
Engineering Society, October 1997.