The Oscillating Torque On A Three-Phase Induction Generator
Connected To A Single-Phase Distribution System
R.G. de Mendonça (MSc), L. Martins Neto (Dr.), J.R. Camacho (PhD)
Electrical Machines Laboratory, Electrical Engineering Department
Universidade Federal de Uberlândia
Campus Santa Mônica, Uberlândia, 38400-902, MG, BRAZIL
e.mail: jrcamacho@ufu.br
Abstract - This paper presents a time domain model for
the three-phase induction generator feeding a single-phase
(two-wire) system. The purpose of this work is to access
oscillation modes inside the induction machine. These
oscillation modes will be impossible to detect when using
frequency domain techniques. A 2 HP three-phase induction
generator is used for the digital simulation. From this study
becomes clear the existence of such torque oscillations.
models[1]. However, the frequency domain technique is not
suitable for the behavioral analysis of an induction
generator concerning torque oscillations inside the machine.
To cover this gap this paper has been designed to present a
mathematical model developed to make possible a time
domain analysis[2],[3],[4]. This model allows this study by
showing the behavior of the oscillating electromagnetic
torque inside the three-phase induction generator.
Index terms - Induction generator, single-phase system,
rural consumers.
I. INTRODUCTION
Countries with such large areas of agricultural land like
Brazil, where electrical energy consumers in rural areas is
characterized by: small consumers in kWh/month; small
quantity of consumers per km of electrical network; small
maximum simultaneous demand of electrical energy, and
limitations on financial resources for investments in
electrification programs. These characteristics lead the
electrical energy authorities to adopt high-voltage singlephase (one wire) and low-voltage (two-wire) electrical
distribution systems. Due to a constant need to increase his
demand for electrical energy, mainly in terms of maximum
demand, the rural consumer can be constrained by
particular characteristics of a single-phase system.
However, in some regions with small hydro-electric
energy availability, it will always be the possibility of
electrical energy generation from small plants with threephase squirrel-cage induction machines. These machines
will be connected to a single-phase distribution system in
order to supply typical three-phase loads.
Figure 1 – Connection diagram for the three-phase squirrel-cage induction
generator. Connected to a single-phase distribution system.
Previous work with a three-phase induction generator
has been done with frequency domain mathematical
Figure 2 – Demonstration of the internal rotating magnetic field in a threephase induction generator connected to a single-phase system.
II. INDUCTION GENERATOR
The proposed three-phase induction generator is
connected to a single phase electrical distribution system. It
is an ordinary induction machine with a squirrel cage rotor
and stator phases displaced spatially by 120o and same
number turns/phase. An adequate capacitance Cap is
coupled between phases B and C, as in Figure 1. The
objective is to obtain a reliable operating point for this
unbalanced configuration, where the generator is delivering
nominal power under nominal voltage. As described in
reference [1], a balanced situation between generator
phases can be obtained, and this balance is of fundamental
importance concerning the connection of a three-phase
induction generator in parallel with a single-phase voltage
source. Through the frequency domain mathematical
modeling proposed in [1], it is clear the good behavior for
the proposed generator, in terms of "rms" values, where the
voltage unbalance is not so accentuated. However, an
interesting point not brought in picture in [1] is the behavior
of electromagnetic torque inside the generator, for small
voltage unbalancing the internal electromagnetic torque
shows a swinging behavior. Those swings are related to the
current unbalance in the generator phases as part of a
unbalanced three-phase system. As a consequence this
generator has different magneto-motrice forces (mmf) in
each phase. This unbalancing allows the decomposition of
the mmf distribution in such a way to obtain static and
rotary magnetic field distribution, where the latter are of
positive and negative sequence. The positive sequence
magnetic field, BR1, under the three-phase symmetrical
induction machine working principles, create indirectly a
rotating magnetic field, BS1, of same sequence, in the
machine's rotor. The same happen to the negative sequence
magnetic field, BR2, creating another BS2. The above is
illustrated in Figure 2, and can be seen clearly that:
- The magnetic fields (BS1, BR1) and (BS2, BR2) shows a
constant angular displacement between them, therefore
the resulting aligning electromagnetic torque is
constant;
- The angular position between magnetic fields (BS1,
BR1) and (BS2, BR2) changes periodically, therefore the
electromagnetic torque is an oscillating value.
The model presented in [1] doesn't allow the possibility
of instantaneous analysis. The referenced model doesn't
allow the analysis of generator internal electromagnetic
torque, therefore in this work it is presented a time domain
mathematical modeling.
III. MATHEMATICAL MODEL
Figure 1 shows the connection diagram for the threephase induction generator connected to a single-phase
distribution system.
From Figure 1 we have:
d(Vb − Vc )
1
.i Cap
=
dt
Cap
ic = −(ia + ib )
V = Va − Vc
Va − Vb = R.i ab
Vb − Vc = R.ibc
Vc − Va = R.ica
(1)
(2)
(3)
(4)
(5)
(6)
The induction generator voltage and current relationship
is presented by the generic equation:
v i = ri .i i +
dλ i
dt
(7)
where:
vi , ii , ri , λi – are respectively voltage, current, resistance
and coupled magnetic flux for one of the generator’s phase
called.
i index – represents one of the generator’s phase (stator abc
– rotor ABC).
The magnetic flux coupling are given by the following
general equation:
λi = (Li + Lii ).i + ∑ Lij .ii
(8)
i≠ j
where:
Li + Lii – generator’s phase i self inductance.
Lij – generator’s mutual inductance between phases i and j.
ii , ij – instantaneous currents of phases i and j ,
respectively.
For the three-phase generator pictured in Figure 1, self
and mutual inductances are given by [L] matrix as follows:
[L1]
[L] =  
[L 2]
(9)
 A11 A12 0 A13 A14 A15 
[L1] =  A12 A22 0 A23 A24 A25 
 0
0 1 0
0
0 
0 
 A13 A23 0 A33 0

[L2] =  A14 A24 0 0 A33 0 
 A15 A25 0 0
0 A33 
(10)
(11)
where:
A11=2(Lss-Mss);
A12= - (Lss+Mss);
A13=Mss-MsR;
A14=Mss-MsR;
A15=0;
A22= -2Mss;
A23=0;
A24=0;
A25=0;
A33=LRR.
Lss - are the values of self inductances for any of the stator
phases (a, b, c);
LRR – are the values of self inductances for any of the rotor
phases (A, B, C) referred to the stator;
Mss – are the values of mutual inductances between stator
phases (a, b, c);
MsR - are the values of mutual inductances between stator
phases (a, b, c) and rotor phases (A, B, C).
Laa=Lbb=Lcc;
Lab=Lac=Lbc;
MaA=MbB=McC=M.cosθ
(12)
MaB= MbC= McA=M.cos(θ - 2π/3)
MaC= MbA= McB=M.cos(θ + 2π/3)
The three-phase induction
equations are given by:
generator
mechanical
dW R 1
= .(Tb − Tm )
dt
J
(13)
dθ p
= .WR
dt
2
(14)
Tm =
p T  ∂L 
.[I ] . .[I ]
4
 ∂θ 
IV. DIGITAL SIMULATION
(15)
where:
WR – rotor angular mechanical speed;
θ - rotor electrical angular displacement;
p – pole number;
Tm – generator electromagnetic torque;
Tb - turbine torque.
Once having all the induction generator equations,
digital simulations have been done in order to confirm the
electromagnetic torque oscillations. Therefore, the
benchmark was the three-phase induction motor used in [1],
a three-phase induction machine with similar characteristics
was taken, a 2 HP, 4 poles, 380/220 Volts, squirrel cage
rotor, with equivalent circuit parameters given by Table 1.
From the union of electrical (1) to (12) and mechanical
(13) to (15) equations, can be obtained an equation system
which represents the three-phase induction generator
connected to a single-phase distribution system. The
resulting matrix equation system is shown below:


d [I ']
 d [L']  
.[I '] (16)
= [L']−1.[V '] −  [R'] + 
dt
 dt   


where:
[I’] - represents the current matrix (stator, rotor and load);
[L’] - represents the inductance matrix (stator, rotor and
load);
[V’] - represents the voltage matrix (stator, rotor and load);
[R’] – represents the resistance matrix (stator, rotor and
load).
TABLE 1. THREE-PHASE INDUCTION MACHINE EQUIVALENT CIRCUIT
PARAMETERS.
Stator Resistance
3,80±0,03Ω
Rotor Resistance (referred to
the stator)
3,01±0,03Ω
Locked Rotor Reactance (
referred to the stator)
Phase Magnetization
Reactance
Figure 4 – No-load induction generator speed, considering only the 1st
spatial harmonic.
The following simulations are obtained with the same
capacitor used in [1], i.e., Cap = 95 µF in order to have an
idea of its effect on the electromagnetic torque behavior
inside the generator.
3,10±0,03Ω
75,15±0,7Ω
Figure 5 – Phase “a” current, considering only the 1st spatial harmonic.
Once simulated the case described in [1], with the
utilization of 95 µF capacitor, through Figures 3, 4, 5, 6 and
7 the electromagnetic torque, speed and generator stator's
phase current time domain graphs. Through Figure 3 can be
observed clearly the presence of oscillating torque around
the turbine's nominal torque (7.5 N.m).
Figure 3 – No load induction generator torque, considering only the 1st
spatial harmonic.
V. CONCLUSIONS
Figure 6 – Phase “b” current, considering only the 1st spatial harmonic.
However, it can be seen in Figure 4, that the generator
speed oscillates expressively. Those oscillations can
produce in a long term basis some stress damage in the
shaft.
From Figure 2 can be observed clearly that the heavy
electromagnetic torque oscillations are the cause of highly
oscillating rotational speed for this machine, as can be seen
in Figure 3. With the use of frequency domain modeling it
is not possible to see clearly these effects. Driving an
induction generator in such conditions it is not advisable
since it’s performance won’t be acceptable. So, it will be
necessary to minimize these oscillations in order to be
possible the use of three-phase induction generators
connected to a single-phase (two-wire) distribution system.
In order to minimize the problem, further studies are in
progress taking advantage of the same philosophy already
used for the asymmetrical three-phase induction motor.
VI. REFERENCES
[1]
T. F. Chan; "Performance Analysis of a Three-Phase
Induction Generator Connected to a Single-phase
Power System.", IEEE – Transactions on Energy
Conversion, September 1998, Vol 13 No. 3, pp 205213.
[2]
Mendonça, R.G.;Martins Neto, L; "Análise
Comparativa de Desempenho: Motor de Indução
Trifásico Simétrico e Motor de Indução Trifásico
Assimétrico. Conjugados Oscilantes", XII Brasilian
Automatic Control Conference – XII CBA. Vol I,
pp. 243-247 – September 14-18, 1998 – Uberlândia,
MG, Brazil.
[3]
Martins Neto, L.; Mendonça, R.G.; Camacho, J.R.
and Salerno, C.H.; "The Asymmetrical Three-Phase
Induction Motor Fed by Single Phase Source:
Comparative Performance Analysis", IEEE-IEMDC
- International Electrical Machines and Drives
Conference, Milwaukee, Wisconsin, May 1997.
[4]
Martins Neto, L.; Camacho, J.R. and Salerno, C.H.
& Alvarenga, B.P., "Analysis of Three-Phase
Induction Machine Inclunding Time and Space
Harmonic Effects: The A,B,C Reference Frame",
IEEE-PES - Transactions on Energy Conversion,
Volume 14, Number 1, pp. 80-85, March 1999.
Figure 7 – Phase “c” current, considering only the 1st spatial harmonic.
It can be observed in Figures 5, 6 and 7 the unbalancing
in the stator windings phase currents. Those are the currents
responsible for the mmf internal unbalance in the generator
and as a consequence the creation of oscillating torque.
From Figure 2 can be seen clearly the oscillating torque
action inside the machine and through Figure 3 can be seen
the damaging effects caused to the machine rotational speed
originated by the internal unbalance inside the machine.