The Asymmetrical Three-Phase Induction Motor Fed by Single Phase ...

The Asymmetrical Three-Phase Induction Motor Fed by Single Phase Source: Comparative Performance Analysis. Carlos H. Salerno(Non-Member, IEEE), José R. Camacho(Member, IEEE) Luciano Martins Neto(Non-Member, IEEE) and Roberlan G. Mendonça (Non-Member, IEEE) Electrical Engineering Department - Universidade Federal de Uberlândia P.O. Box 593 , 38400-902 - Uberlândia - MG - Brazil Abstract: 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 induction motor to produce nominal power at nominal speed. This paper will present the following structure: discussion of the mathematical modeling, study for the minimization of the oscillating torque, starting time, and etc..., 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.

function of the motor load. However some additional problems as mentioned in reference [1], need to be solved. One of the problems is the undesirable effects in the oscillating torque. The objective of this paper is to investigate dynamically the oscillating torque in search for its minimization. Using a mathematical modeling of the asymmetrical motor dynamic operating conditions will be possible to know the source of the oscillating torque and their precise quantitative evaluation. II. MACHINE MATHEMATICAL MODELING

Figure 1 shows that the capacitor current is ib, and from this figure can be derived also the expression of the capacitance voltage in respect to time, which is given by:

dV cap

Keywords: Asymmetrical induction motor, machine modeling, unbalancing factor. I. INTRODUCTION

The asymmetrical three-phase induction motor present some problems when keeping the capacitor in Figure 1 at a fixed value, and it shows also a poorer performance when compared with the ordinary three-phase symmetrical induction motor fed by a three-phase balanced source. To solve some of these problems, has been suggested in the past the possibility to change dynamically the value of Cap as a

dt

=−

1 . ib Cap

(1)

with also the circuit analysis of Figure 1 the two following equations can be obtained: i c = − (i a + i b ) (2)

V = Va − Vc

(3)

Figure 1. Schematic diagram of an asymmetrical three-phase induction motor.

A 6x6 impedance matrix [L] should represent the majority of aspects of the asymmetrical motor, such as the different turns ratio in each phase, the 1200 between the phase

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windings A, B and C in the stator and their different number of turns making the windings asymmetrical. The flux linkage [λ] and current [I] vectors are represented by two 6x1 vectors. So the expression for the flux linkage is: (4) [λ ] = [ L ].[I ] Induction motor voltages may be represented by:

v = r. i +

dλ dt

horse-power of the desired asymmetrical three-phase induction motor. The asymmetrical motor obtained from a symmetrical motor has the same capability, with modifications only in number of coil turns in each phase. In this case phase a is assumed fixed in terms of number of turns, and even with this constraint it will be possible to obtain a large number of variations in the other two phases number of turns combined with the applied capacitor. Was observed that modifications around the optimum doesn’t change much the motor performance with respect to speed, torque, and power. With different number of turns per phase our motor is now unbalanced with respect to the magnetomotive forces in each phase, a compensation should be made in the machine unbalancing factor (F) in order to make it as close to zero as possible[3]. It is known that the machine unbalancing factor is a function of “b”, “c” and “Cap”. So, to make our machine design with the lowest possible unbalancing rate without changing its torque, power and speed conditions, it is necessary to make a computational study in order to find which combinations of “Cap”, “b” and “c” can take the unbalancing factor to zero. Once obtained these values, another computational analysis should be made to obtain currents, torque in per unit, unbalancing factor, etc... An important detail that is worth to mention, is that for some values of “b”, “c” and “Cap” the better torque condition is not always when F=0. With F having a tendency to go to zero, the average torque in per unit also have a tendency to be a lot smaller than unit. This condition is not desirable in our case, an optimization study should be made, and to have the torque close to the unit it is necessary to have a small unbalancing between MMFs. Reference[3] can show that the searched solution is not unique, and a range of combinations that would satisfy our main conditions can be obtained for our 2 HP three-phase induction motor.

(5)

where i(a, b, c, A, B, C) and r(a, b, c, A, B, C) are the stator and rotor currents and the stator and rotor resistances respectively. The differential equation for flux can be obtained from equations (5) and (1). The mechanical equations for the motor are given by:

dw R 1 = .(Tm − Tl ) dt J dθ p = . wR dt 2 ∂L p Tm = [ I ]T .[ ].[ I ] 4 ∂θ

(6) (7) (8)

To have reasonable simulating conditions, parameters wR and Tm will be used as state variables, so equations (6) and (8) should be expanded in such a way that the derivative of rotational speed could be obtained as a function of currents and displacement angle (θ). Using equation (8) with the motor dynamic equations, the system state equations can be arranged as:

d[ I b ] = [V , ] − [ R , ].[ I , ] dt

(9)

with:

[ I b ] = [ L, ].[ I , ]

(10)

Consequently:

d [ I b ] d [ L, ] , d[ I , ] = .[ I ] + [ L, ]. dt dt dt

IV. THE UNBALANCING FACTOR

(11)

Taking into consideration the stator in Figure 1 for the asymmetrical induction motor, Y connected, fed by a singlephase source between phases A and C and with an auxiliary capacitor (Cap) between phases B and C. This motor will be working under the influence of two rotating magnetic fields which rotate in opposition to each other, as a consequence each of these magnetic fields will produce torques that have also opposite direction. One parameter that gives a relation between these torques is the relation between positive and negative sequence MMF’s. This relation is nominated as unbalancing factor (F). That will be our main guide to choose (Cap) in order to minimize torque oscillations.

and:

[V , ] − [ R , ].[ I , ] =

d [ L, ] , d[ I , ] .[ I ] + [ L, ]. dt dt

(12)

Thus the solution for the problem will be given by applying the following equation:

d[ I , ] d [ L, ] , −1 , , = [ L ] ([V ] − ([ R ] + ).[ I , ]) dt dt

(13)

where the derivatives in respect to t should be changed to θ. III. OBTAINING AN ASYMMETRICAL INDUCTION MOTOR

For the design of an asymmetrical three-phase induction motor with single-phase feeding it is necessary, before anything, to obtain the parameters for the equivalent circuit of a three-phase symmetrical induction motor with the same

F& =

2

& a2 mmf & a1 mmf

(14)

V. BALANCED AND UNBALANCED MOTOR TORQUE RATIO

Since:

& a1 = N a I&ap mmf & a = N a I&an mmf

(15)

The power delivered to the rotor in each phase is the difference between the power delivered to the rotor due the positive and negative sequence impedances. So developing all those equations the asymmetrical induction motor torque final equation is given by:

(16)

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substituting equations (15) and (16) in equation (14) we have the unbalancing factor as a ratio between the positive and negative sequence from one of the phase currents, phase A in this case. Our objective now is to make this factor (F) a function of the motor parameters, Cap and the turns ratio between phases B and C called b and c. Using the negative, positive and zero sequence circuit equations for the conventional three-phase induction motor with peculiarities shown in Figure 1. The unbalancing factor is given by:

T=

3V 2 [Re( Z& \ ap ) − Re( Z& \ an ). F 2 ] ws Z2

where Re(Z) is the real part of Z impedance and Z’ represents rotor impedances referred to the primary. Balanced torque Tb is defined in our case as being the torque delivered by a three-phase symmetrical induction motor (b = c = 1), being fed by a three-phase voltage balanced system. The balanced torque equation will be given in its simplest form by:

I&an Z&1 − Z& 0 .α 1 + j. b. Xc.(α 1 − α 2 ) & F= = (17) I&ap Z& 0 .α 2 − Z& 2 + j. b. Xc.(α − α 2 ) where:

α = cos 120 0 + j. sen 120 0 α1 =

1 + b. α + c . α 1+ b + c

α2 =

1 + b. α + c. α 1+ b + c

Tb =

2

2

t=

are the phase operators and:

Z&1 Z& 2

Re( Z& \ ap ) V 2 .[ ] ws Zap

(20)

A ratio between unbalanced and balanced torques (t) is obtained dividing equation (19) and (20), in this case we have:

(18)

Z& 0

(19)

2 T 3Zap Re( Z& \ an ) 2 .F ] = 2 [1 − Tb Z Re( Z& \ ap )

(21)

VI. PROTOTYPE TEST DATA AND SIMULATION RESULTS

1 1 = ( − ). Z& sa b c α2 α & =( − ). Z ap b c α α2 & =( − ). Z an b c

From tests in three phase 2 HP symmetrical and asymmetrical induction motors, parameters were obtained to I.M. data Power (HP) Voltage (V) Current (A) Frequency(Hz) Speed (rpm) Torque (N.m) Is/In Insulation Class Power Factor Service Factor

where Zsa, Zap and Zan are respectively zero, positive and negative sequence impedances of the original induction motor, obtained from the induction motor equivalent circuits, while, Z0, Z1 and Z2 are respectively the same impedances for the asymmetrical induction motor. It is an important observation that all the electrical machine parameters are known and the unbalancing factor will be function only of b, c and Cap. Once obtained those equations developed for the asymmetrical induction motor case, phase currents, line and phase voltages will be obtained from circuit equations. In this case can be determined also the current, power and power factor for the two phase feeder that is the source for the asymmetrical motor. So the torque can be obtained through the power delivered in the rotor times the synchronous speed (ws).

Asymmetrical 2.0 380 5.0 60 1720 7.3 B -

Symmetrical 2.0 380/220 3.99/6.99 60 1720 7.9 6.8 B 0.78 1.15

Table I - Induction motor data.

The symmetrical induction motor shows the following equivalent circuit parameters: Xs (Ohms) 3.10 ± 0.03 Xm (Ohms) 72.15 ± 0.7 Xr (Ohms) 3.10 ± 0.03 Rs (Ohms) 3.80 ± 0.03 Table II - Equivalent circuit parameters in 60 Hz for the symmetrical induction motor.

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be used as data for digital simulation. According to a previous and cumbersome work done by Dr. Martins Neto[3] the optimum values found for the 2 HP, 4 poles, motor used in this study were: - case 1: b = 0.6, c = 1.4 and Cap = 40 µF; - case 2: b = 0.6, c = 1.8 and Cap = 37 µF. A. Digital Simulation This simulation is based on the data and mathematical formulation presented above. Its results will give further insight in the analysis of oscillating torque.

Figure 4 - Induction motor torque for the second case.

Figure 2 - Symmetrical induction motor torque

Peculiarities of the asymmetrical machine that are responsible for originating this torque are studied, such as the combination of unbalancing factor, winding turns ratio and capacitance, in a search for the minimization of the unbalanced torque. Figures 3 and 5 show the torque for the first data combination case and Figures 4 and 6 for the second case. Figures 5 and 6 were zoomed to show oscillations in detail.

Figure 5 - Zoomed induction motor torque for the first case.

Figure 6 - Zoomed induction motor torque for the second case.

B. Laboratory Tests.

Figure 3 - Induction motor torque for the first case.

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the average torque of 7.58 Newton.metres and torque oscillation of about 4%. For the first laboratory prototype test the average nominal torque shown in Figure 8 was 6.97 [N.m], 8% lower than in the three-phase symmetrical induction motor test, and shows a speed of 1733 rpm.

An objective of this work was to design and build an asymmetrical induction motor fed by a single-phase (two wire) source. This motor should give the nominal torque with a low presence of oscillating modes and close to the nominal torque of the same rating symmetrical motor.

Figure 7 - Symmetrical induction motor torque at rated load.

Figure 9 - Asymmetrical induction motor torque for the second laboratory test.

In the second laboratory prototype test case the average nominal torque shown in Figure 9 is 6.85 [N.m], 9.6% lower than in the three-phase symmetrical induction motor test, and shows a speed of 1687 rpm. VII. CONCLUSIONS

It is worth mentioning that the combination of parameters which gives a better performance for the asymmetrical threephase induction motor is not simple. Based in this complexity we felt challenged to obtain an asymmetrical induction motor which performance was very much like the symmetrical three-phase induction motor. The unbalancing factor, as mentioned before, is not necessarily equal zero in the case of balanced and unbalanced torque ratio equal unity. This is a result from the fact that the T/Tb ratio depend upon the optimum combination of values for b, c and Cap. Optimum values for this ratio can be selected for a range of induction motor sizes based in a case by case study. Figures 2 and 3, obtained through computer simulation, shows respectively peak to peak oscillations in steady-state of 1.17 N.m and 0.62 N.m. It can also be observed that the starting time is almost the same in both cases. These graphics show clearly the efficiency of proposed balancing method for the asymmetrical induction motor fed by single-phase (two wire) source. An interesting observation that can be made in the machine behavior when changing c from 1.4 to 1.8, is the low sensitivity of torque in respect to phase number of windings. The steady-state torque, comparing zoomed Figures 5 and 6 for both cases, has no noticeable changes in

Figure 8 - Asymmetrical induction motor torque for the first laboratory test.

After the prototype was built, information was obtained through a data acquisition system. The two following tests were done as close as possible to the digital simulation: - first test: b = 0.6, c = 1.4, starting Cap = 71.67 µF and running Cap = 39.58 µF, - second test: b = 0.6, c = 1.8, starting Cap = 76.71 µF and running Cap = 36.95 µF. Both tests were compared with the test for the three-phase symmetrical induction motor with around of 2% of forced unbalancing between stator voltages. Figure 7 shows the torque for the symmetrical induction motor at rated load and

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waveshape and average value in Newton.metres, with change from 7.6 to 7.4 which is a 2.6% approximate decrease when increasing c by 28.6%. In the case of changing c of that amount the starting torque had higher peaks but the starting time remained almost the same. Currents in phases a, b, c in the stator are little more balanced with c=1.4, but they still show the characteristic unbalancing with Ic always being the smallest. However the speed is less uniform in steady-state for this case. Torque results from laboratory tests show, when selecting an optimum combination of parameters, that the torque oscillation is always lower than 10%. In Figures 7 and 8 can be observed that torque oscillations in the asymmetrical motor can be satisfactory, since that cases show oscillations around 6%. Important conclusions are: - the torque sensitivity, due the asymmetries, with the change of capacitance in steady-state, the average torque changes with the capacitance; - power factor of both tested versions were close enough to unity (0.974 and 0.938 respectively) at nominal load, becoming an advantage factor when compared with single phase induction motors; - the proposed asymmetrical induction motor has no need for starting gadgets, starting schemes are responsible for the majority of technical problems in single-phase induction motors. The above observed facts suggest that in the case of asymmetrical induction motor the changing in design characteristics is very much an optimization problem. It will be also technically possible to build asymmetrical induction motors larger than 10 HP.

Brazilian Commitee of CIGRÉ-JWG 11/14-09 (Unit Connected Generation). His areas of interest are Dynamic Simulation, Electrical Machines Simulation and High Voltage AC-DC conversion. Luciano Martins Neto - Dr. Martins Neto was born in Botucatu, SP, Brazil in 22/05/48. He has a Doctoral degree in Electrical 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. Roberlan G. de 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. Currently, he is doing his Master’s degree from Universidade Federal de Uberlândia, his area of interest is Electrical Machines. IX. REFERENCES [1]

[2]

[3]

VIII. BIOGRAPHY [4]

Carlos Henrique Salerno - Dr. Salerno was born in Uberlândia, MG, Brazil in 31/05/61. Completed his Doctoral degree at UNICAMP - Universidade de Campinas Faculdade de Engenharia Elétrica - São Paulo - Brazil in December 1992. He is a Senior Lecturer in the Electrical Engineering Department at Universidade Federal de Uberlândia, MG, Brazil where he works since January 1992. His areas of interest are Electrical Machines and Dynamic Simulation.

[5]

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 Researcher-Consultant of CNPq (Brazilian National Council for Scientific and Technological Development) and collaborator-member of 6

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. Tozune, A., Balanced Operation of Three-Phase Induction Motor with Asymmetrical Stator Windings Connected to Single-Phase Supply System; IEE Proceedings-B, Vol. 138, n1 4, pp. 167-174, July 1991, London, UK. Martins Neto, L., Three-Phase Asymmetrical Induction Motor; The First International Conference on Power Distribution, Belo Horizonte, MG, Brazil, 1990. In Portuguese. Tindall, C.E. & Monteith, W., Balanced Operation of Three-Phase Induction Motors Connected to Single-Phase Supplies; Proceedings IEE, Vol. 123, n1 6, pp. 516-522, June 1976, London, UK. Straughten, A. & Tracy G.F., Single-Phase Operation of Three-Phase Induction Motors, Engineering Journal, pp. 14-17, February 1969.