Vector Control Strategies for Single-Phase Induction Motor Drive ...

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 5, OCTOBER 2004

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Vector Control Strategies for Single-Phase Induction Motor Drive Systems Maurício Beltrão de Rossiter Corrêa, Member, IEEE, Cursino Brandão Jacobina, Senior Member, IEEE, Edison Roberto Cabral da Silva, Fellow, IEEE, and Antonio Marcus Nogueira Lima, Member, IEEE

Abstract—This paper discusses vector control strategies for single-phase motor drive systems operating with two windings. A model is proposed and used to derive control laws for single-phase motor drive systems. Such model is also employed to introduce the double-sequence controller. Simulation and experimental results are provided to illustrate the operation of the proposed drive systems. Index Terms—Single-phase induction motor drive, vector control. Fig. 1. Configuration of the single-phase drive system.

I. INTRODUCTION

S

INGLE-PHASE machines are widely employed for low-power applications. In those applications the machine operates at constant frequency and is fed directly from the ac grid without any type of control strategy. The single-phase machine has both a main winding and an auxiliary winding and its operation requires one or two capacitors (startup and running capacitor). The cost reduction of the semiconductor switches and the need to provide power-factor control to guarantee the efficient use of energy, even for low-power applications, has stimulated the investigation of different single-phase motor drive schemes [1]–[6]. In these schemes the single-phase machine, without its startup and running capacitors, is treated as an asymmetric two-phase machine. The development of a high-performance low-cost single-phase motor drive system is a very useful endeavor. Such a system provides high-quality electromagnetic torque to the load as well as offering the possibility of variable-frequency operation, which can be exploited to improve the efficiency of the application. Different static converter topologies have been used to supply the single-phase machine, for example, [2], [6] and [7]. [2] and [6] propose configurations employing four switches. In [2] a converter is used for emulating a variable capacitor but without power factor control. In [6] switches are used to obtain a twophase inverter but the input rectifier is not considered. That paper also considers the use of the topology proposed in [5] and [7] but the focus given here is on the detailed analysis and design of the vector control strategies for single-phase motor drives. The basic drive configuration being studied in this paper is presented in Fig. 1. This configuration is indicated for high-per-

formance applications and provides power-factor control and the possibility of operation in regenerating mode. In a previous paper the authors presented the rotor-flux vector control strategy for single-phase motor drives [8]. In this paper the previous analysis is revised and extended to incorporate strategies based on the stator-flux. In particular, some stator-flux vector control strategies (like field-oriented control and stator-flux slip control) have been reformulated to be used with a single-phase machine. Also, a double-sequence controller is introduced to control the stator current. The approach presented here is recommended for applications that require improvement of the overall performance of a single-phase drive system when some mechanical, structural, or economical constraints dictate the use of the already existing machine. Simulation and experimentally obtained results are presented to demonstrate the main characteristics of the proposed drive systems. II. MACHINE MODEL The single-phase machine considered in this paper is magnetically symmetrical and its two windings are displaced so that rad away from the auxiliary the main winding is located one (see Fig. 1). Neglecting the core saturation, the dynamic behavior of the single-phase machine can be described by the following equations: (1)

(2) Manuscript received May 22, 2002; revised December 11, 2003. Abstract published on the Internet July 15, 2004. The authors are with the Departamento de Engenharia Elétrica, Universidade Federal de Campina Grande, 58109-970 Campina Grande, Brazil (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TIE.2004.834973 0278-0046/04$20.00 © 2004 IEEE

(3)

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(4)

(11)

(5)

(12)

(6) These equations were derived as for an asymmetric two-phase machine [9]. The superscript in the variables of (1)–(6) indicates that the stator reference frame has been adopted. Variables , , , , , , , , , and are voltages, currents, and fluxes of the stator and rotor in the stator , , and denote the stator reference frame, respectively; , , , , and denote the and rotor resistances; stator, the rotor self-, and mutual inductances; , , and are the machine speed, the electromagnetic torque, and the load torque, in this order; and , , and are the machine pole pairs, the moment of inertia, and viscous friction coefficient, respectively. Note that (1)–(5) are general equations for the two-phase machine. In other words, they may represent either a symmet; ; ) or an asymmetrical rical ( ; ; ) machine. Since the ma( chine studied is asymmetrical, it can be seen from (5) that the machine produces torque oscillations if the and currents are balanced. The term balanced in this paper denotes that the variables and are sinusoidally phase shifted by 90 and with the same amplitude. Rewriting the model (1)–(5) by eliminating some asymmetries is very useful for further vector applications. As it was done in [8] to derive the symmetrical model, also, here, the mutual and will be employed to define a transinductances formation for the stator variables. Such transformation is given by (7)

(13) If the unbalancing depends only on the number of turns of each winding and considering that the number of turns of the main winding is and that the number of turns of the auxilthen the ratio will be approxiiary winding is mately equal to , i.e., . Thus, the transformation expressed in (7) corresponds approximately to refer the auxiliary winding variables to the main winding. Evaluation of and (10), (12), and (13) shows that when the stator currents are balanced, then rotor flux and as well as rotor and are also balanced. Therefore, the machine currents does not produce torque oscillations. As a consequence the magnetizing flux is also balanced. On the other hand, the presence in (9) and in (11) reveals that of and are balanced the and stator voltages and when , then it can be shown that flux are not balanced. If corresponds to the difference between the leakage of -axis stator winding referred to the axis and the leakage of the -axis stator winding. If these inductances are slightly will be almost zero and, thus, (11) will different, provide relatively balanced stator fluxes. With respect to (9) it and must be noted that besides the influence of stator flux, are unbalanced due to . It should be noted that the influence of is relatively reduced at high speed but cannot be ignored at medium or low speed. Note that, differently from the model given by (1)–(5), (9)–(13) are useful in deriving the vector model and vector control strategies, as shown later in this paper. III. ROTOR-FLUX CONTROL

with

and must be applied as follows:

As for any induction motor, to define a control strategy for a single-phase motor (asymmetrical two-phase induction motor) based on rotor-flux orientation requires the definition of relationship among torque, rotor flux, and stator current. Then, by using (12) it is possible to rewrite (13) as follows: (8)

The new mathematical model is

(14) The relationship between the rotor flux and the stator current can be determined by using (10) and (12), yielding the following expression:

(9)

(10)

(15) . where The vector model is defined from (15) by rearranging the variables in the vector form. If this vector model is written for an

DE ROSSITER CORRÊA et al.: VECTOR CONTROL STRATEGIES FOR SINGLE-PHASE INDUCTION MOTOR DRIVE SYSTEMS

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IV. STATOR-FLUX CONTROL Calculation of electromagnetic torque as a function of stator flux and stator currents can be achieved by using (11) and (13), and it can be written as (22)

Fig. 2.

Block diagram of the indirect rotor-field-oriented control.

arbitrary reference frame (denoted by superscript ), which is rad away from phase of the stator, then (16)

with . Since having balanced stator currents is a condition for generating torque free of oscillations it can be seen from (22) that the stator flux of and windings equal to zero. This must be unbalanced enough to make analysis is in accordance with (11), from which it is possible to conclude that balanced currents generate unbalanced stator is null the stator flux fluxes. On the other hand, if is balanced and the torque will not exhibit oscillations. To determine the dynamic model that relates the stator fluxes to the stator currents it is required to perform some algebraic manipulations with (10)–(12). The resulting equations are

where is the speed of the arbitrary reference frame. The variables in the arbitrary reference frame are calculated from the variables in the stator reference frame through the following expressions:

(23)

(17)

(24)

(18) Based on the vector model given by (14) and (16) it is possible to apply the field-oriented principles to control the rotor flux and the electromagnetic torque of the single-phase machine. For that, the rotor-flux reference frame (denoted by the superand . script ) is chosen and, consequently, The torque and flux-current equations in the rotor-flux reference frame can be obtained from (16), that is,

where and . As has been done for the case of the rotor-flux control, the vector model written for an arbitrary coordinate system can be derived from (23) and (24). The vector model for the stator-flux control is given by

(25) (19) (20)

in which

and

(21) with

. Then, controls the rotor flux and controls the electromagnetic torque. As an example, Fig. 2 shows the block diagram of the indirect rotor-field-oriented control scheme, which has been adapted for the single-phase maand represent the reference elecchine. In this diagram tromagnetic torque and amplitude of the rotor flux, respectively. performs the coordinate transformation from the Block reference frame aligned along with the rotor-flux vector to the represents the speed constationary reference frame. Block troller. Furthermore, and represent the reference currents supplied to the stator current controllers, which must be imposed on the machine windings. Block CC VSI IM represents the current controller, the voltage-source inverter, and the induction machine. The analysis of the current controllers will be presented in Section V.

(26) The model in (25) presents an additional term that represents the asymmetry of the machine. Note that this term de, which, as mentioned before, can be pends on quite small. A. Direct Field-Oriented Control Based on the vector model given by (25) and the torque given by (22), it is possible to apply the field-oriented principles to control the stator flux of the single-phase machine. For that, the stator flux reference frame (denoted by superscript ) is chosen and . In this synand, as a consequence, chronous reference frame the torque and stator-flux current can be calculated by (27)

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Fig. 3.


Block diagram of the direct stator-field-oriented control.

Fig. 4. Block diagram of the stator-flux slip control.

(28)

and Neglecting be derived, that is,

a slip-dependent torque expression can

(30)

(29)

Considering that as well as and are negligible, the model becomes symmetric and the conventional stator-field-oriand ented control strategy can be used. In this case, specify the flux and the torque, respectively. Note that, differently from the rotor-flux control, the stator flux depends on and , then, it can be shown that the -axis rotor both current is proportional to . Fig. 3 shows the block diagram of the direct stator-field-oriented control scheme that has been and adapted for the single-phase machine. In this diagram represent the desired electromagnetic torque and amplitude and represent the of the stator flux, respectively. Blocks flux and speed controllers. The amplitude of the stator flux is given by and the angle of the ref. This erence frame is determined by angle is included in the block to perform the coordinate transformation from the reference frame aligned along with the stator flux vector to the stationary reference frame. The stator and flux components are determined by integrating . The performance of the control scheme sketched in Fig. 3 can be improved if a compensation of the disturbing terms in (28) is provided. However, in this work, it was consid. As mentioned in the case of the ered that rotor-flux control, and represent the reference currents supplied to the stator current controllers.

B. Stator-Flux Slip Control The adaptation of the stator-flux slip control strategy for the single-phase machine is illustrated by the block diagram shown in Fig. 4, in which the stator flux loop is closed around the comand . In this case the dypensated flux components namic equations that relate the stator-flux components to the stator voltages are given by (1). The electromagnetic torque control as a function of the slip can be derived from (22) and (25).

where is the machine slip. For the usual slip operation (low slip) (30) becomes (31) The controllers indicated in Fig. 4 are implemented as proportional–integral (PI) controllers and can be designed based on (9). Analog circuitry with an hysteresis approach can also be used for their implementation. Note that the standard Volts/Hertz scheme can also be adapted to be applied to a single-phase machine. In this case, it is reand by the factor quired to compensate the amplitude of , that is, (32)

V. STATOR-CURRENT CONTROL The stator-current control loop is required for the control strategies illustrated in Figs. 2 and 3. The stator-current controller can be implemented by using either an analog hysteresisbased controller or a linear controller [8]. To design a stator-current controller it is necessary to determine the dynamic equation that relates the stator currents to the stator voltages. In this paper, the dynamic model which is employed to design the current controllers is derived from (9)–(12) and is given by

(33)

(34)


The counterelectromotive forces

and

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are given by (35) (36)

and . Therefore, the following vector model can be written in an arbitrary reference frame:

(37) where Fig. 5. Block diagram of the current double-sequence controller.

(43) (44)

In terms of the vector model given by (37), the machine asymmetry is represented by the term . In steady state, is consti(direct sequence) tuted by two vectors with frequencies (inverse sequence), where is the angular stator and frequency. When the machine is symmetrical and a PI controller is employed, the use of the synchronous reference frame has proved to be effective because the disturbance terms are transformed into dc quantities that are easily compensated by the controller itself. In the case of the single-phase machine the use only solves the of the synchronous reference frame disturbance rejection for the positive-sequence term that rotates and after the coordinate transformation at the frequency becomes a dc component. The negative-sequence term, after the coordinate transformation, becomes a component that rotates at and, consequently, cannot be compensated by a single positive synchronous controller. Previous analysis has shown that in the case of the singlephase machine it is necessary to investigate new controller structures that provide better disturbance rejection properties. The controller structure used in this paper employs two different synchronous controllers. The positive-sequence synchronous conand is designed to compensate the directtroller rotates at sequence term. The negative-sequence synchronous controller and is designed to compensate the inverse-serotates at quence term. The individual controllers of this double-sequence controller operate simultaneously and their outputs are added. The control law for the double-sequence (positive–negative sequence) controller can be described as

where is the stationary current error; and are state variables associated to the positive and negative , integral part of the controller; , and are the positive, neg, ative, and stationary reference voltages, respectively; and , , and are the gains for the positive and negative part of the controller, respectively. In these notations superscript , and indicate the reference variable, positive and negative reference frames, respectively. Fig. 5 shows the block diagram of the stator current control and indicated in this loop. The input reference currents diagram are obtained at the outputs of block diagrams of Figs. 2 and 3. The outputs of the current controller provide the modulating waveforms for the pulsewidth modulator. The computational load of a given control law can be estimated by the number of additions and multiplications required to implement it. Compared to the positive-sequence synchronous controller presented in [8], the double-sequence controller almost doubles the computational load. However, if there is no restriction in terms of processing power, the double-sequence controller can provide better performance than the positive-sequence controller. The implementation of the double-sequence controller can be simplified if it is emulated in the stationary reference frame, and , as proposed in [10] avoiding the transformations for the positive synchronous controller. By introducing , , and and by using (38)–(44), the equations of the continuous-time control law of the double-sequence stationary controller can be rewritten as (45)

(38) (39) (40) (41) (42)

(46) (47) The use of the same gains and simplifies even more the control law implementation. In fact, from (45)–(47) for and introducing the new variables

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Fig. 6. Rotor flux and speed, reference (dashed–dotted line) and actual (solid line), waveforms (scheme of Fig. 2).

Fig. 7. Stator flux and speed, reference (dashed–dotted line) and actual (solid line), waveforms (scheme of Fig. 3).

and the equations of the double-sequence stationary controller become (48) (49) (50)

VI. SIMULATION RESULTS All control strategies were studied by simulations. Selected results are presented in Figs. 6–8. These results were obtained with a machine with and fed by an ideal voltage source. A PI controller was used as the current controller in the strategies of Figs. 2 and 3. The flux controller used in Fig. 3 was also a PI controller. Both controllers in Fig. 4 were of the PI type. Both the rotor-flux reference (Fig. 6) and the stator-flux reference (Figs. 7 and 8) are constant and equal to 0.4 Wb. The reference for the mechanical speed is given by rad s for

s

rad s for

s

s

rad s for s rad s for

s s

s

rad s for s rad s for

s s

Figs. 6–8 show the transient waveforms observed when the machine starts from standstill and is controlled with the strategies of Figs. 2–4, respectively. It can be noted that the results obtained are satisfactory. Absence of disturbance in the rotor-flux model results in a more precise control of rotor flux when compared with the stator-flux approach. The current control strategy that employs the double-sequence controller (Fig. 5) has been tested by simulation and the results were satisfactory. Fig. 9 shows the waveforms of for the double-sequence and the positive synchronous

Fig. 8. Stator flux and speed, reference (dashed–dotted line) and actual (solid line), waveforms (scheme of Fig. 4).

controllers for step changes ( s and s) of the reference currents. As can be noticed, the disturbance rejection was improved with respect to the strategy that utilizes a positive synchronous controller. However, it must be remarked that for that machine the use of a positive synchronous controller provides a very acceptable reduction of the disturbance terms . VII. EXPERIMENTAL RESULTS In order to corroborate the previously proposed scheme an experimental prototype with the same standard single-phase machine of the simulation study was used. The control of the inverter was realized by means of a computer with a dedicated plug-in board that provides the interface and data acquisition functions. As an example, the speed control of the machine was implemented according to the block diagrams of Figs. 2 and 4. The speed reference profile was the same as that used for the simuand lation study. Fig. 10 presents the results for the actual and the reference and variables when the machine operates with the stator-flux slip control strategy. For the results shown in Fig. 11 the machine operates with the indirect rotor-field-oriented control strategy and the actual , . and


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As can be observed the operation of both schemes is satisfactory, but the speed control at the start is superior in the case of the rotor-flux control.

VIII. CONCLUSION

Fig. 9. Stator current error: positive synchronous controller (top) and double-sequence controller (bottom).

Fig. 10. Experimental stator flux and speed, reference (dashed–dotted line) and actual (solid line), waveforms (scheme of Fig. 4).

Fig. 11. Experimental stator currents and speed, reference (dashed–dotted line) and actual (solid line), waveforms (scheme of Fig. 2).

and the reference

,

, and

variables are presented.

This paper has discussed some vector control strategies for single-phase motor drive systems operating with two windings. The modeling approach proposed made it possible to adapt some high-performance control strategies for use with a single-phase motor drive system. This approach also provided a simple representation of the single-phase machine asymmetry, which is very useful to understand some features of a single-phase drive system. Experimental tests were considered satisfactory and have confirmed the claimed features. In spite of being more complex than other strategies, like the standard V/Hz, the use of field-oriented control may not result in additional cost in terms of data processing. This is due to the constant innovations of microelectronics that increase the capability of processing information and reducing cost and power consumption of the integrated circuits. In perspective, the techniques presented here—concerning the machine model and current controllers for single-phase machine system (asymmetrical two-phase machine system)—can also be extended to improve functionality of three-phase drives submitted to certain types of faults. For example, there is the case in which an internal short circuit significantly reduces the number of turns in one of the machine windings so that the machine becomes unbalanced.

REFERENCES [1] E. R. C. Jr, A. B. Puttgen, and W. E. S. II, “Single-phase induction motor adjustable speed drive: Direct phase angle control of the auxiliary winding supply,” in Conf. Rec. IEEE-IAS Annu. Meeting, 1988, pp. 246–252. [2] T. A. Lettenmaier, D. W. Novotny, and T. A. Lipo, “Single-phase induction motor with an electronically controlled capacitor,” IEEE Trans. Ind. Applicat., vol. 27, pp. 38–43, Jan./Feb. 1991. [3] D. G. Holmes and A. Kotsopoulos, “Variable speed control of single and two phase induction motors using a three phase voltage source inverter,” in Conf. Rec. IEEE-IAS Annu. Meeting, 1993, pp. 613–620. [4] M. F. Rahman, L. Zhong, and S. Y. R. Hui, “A single-phase, regenerative, variable speed induction motor drive with sinusoidal input current,” in Conf. Rec. EPE’95, 1995, pp. 3777–3780. [5] M. F. Rahman and L. Zhong, “A current-forced reversible rectifier fed single-phase variable speed induction motor drive,” in Proc. IEEE PESC’96, 1996, pp. 114–119. [6] C. M. Young, C. C. Liu, and C. H. Liu, “New inverter-driven design and control method for two-phase induction motor drives,” Proc. IEE—Elect. Power Applicat., vol. 143, pp. 458–466, Nov. 1996. [7] P. N. Enjeti and A. Rahman, “A new single-phase to three-phase converter with active input current shaping for low cost ac motor drives,” in Conf. Rec. IEEE-IAS Annu. Meeting, 1990, pp. 935–939. [8] M. B. R. Corrêa, C. B. Jacobina, A. M. N. Lima, and E. R. C. da Silva, “Rotor-flux-oriented control of a single-phase induction motor drive,” IEEE Trans. Ind. Electron., vol. 47, pp. 832–841, Aug. 2000. [9] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Machinery. Piscataway, NJ: IEEE Press, 1995. [10] T. M. Rowan and R. J. Kerkman, “A new synchronous current regulator and an analysis of a current-regulated pwm inverter,” IEEE Trans. Ind. Applicat., vol. 22, pp. 678–690, July/Aug. 1986.

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Maurício Beltrão de Rossiter Corrêa (S’97–M’03) was born in Maceió, Brazil, in 1973. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from the Federal University of Paraíba, Campina Grande, Brazil, in 1996, 1997, and 2002, respectively. From 1997 to May 2004, he was with the Coordenação de Ensino Tecnológico-Centro Federal de Educação Tecnológica de Alagoas, Palmeira dos Indios, Brazil. Since June 2004, he has been with the Electrical Engineering Department, Federal University of Campina Grande, Campina Grande, Brazil, where he is currently an Assistant Professor of Electrical Engineering. From 2001 to 2002, he was with WEMPEC, University of Wisconsin, Madison, as a Scholar. His research interests include power electronics and electrical drives.

Cursino Brandão Jacobina (S’78–M’78–SM’98) was born in Correntes, Brazil, in 1955. He received the B.S. degree in electrical engineering from the Federal University of Paraíba, Campina Grande, Brazil, in 1978, and the Diplôme d’Etudes Approfondies and the Ph.D. degree from the Institut National Polytechnique de Toulouse, Toulouse, France, in 1980 and 1983, respectively. From 1978 to March 2002, he was with the Electrical Engineering Department, Federal University of Paraíba. Since April 2002, he has been with the Electrical Engineering Department, Federal University of Campina Grande, Campina Grande, Brazil, where he is currently a Professor of Electrical Engineering. His research interests include electrical drives, power electronics, control systems, and system identification.

Edison Roberto Cabral da Silva (SM’95–F’03) was born in Pelotas, Brazil, in 1942. He received the B.C.E.E. degree from the Polytechnic School of Pernambuco, Recife, Brazil, in 1965, the M.S.E.E. degree from the University of Rio de Janeiro, Rio de Janeiro, Brazil, in 1968, and the D. Eng. degree from the University Paul Sabatier, Toulouse, France, in 1972. From 1967 to March 2002 he was with the Electrical Engineering Department, Federal University of Paraíba, Campina Grande, Brazil. Since April 2002, he has been with the Electrical Engineering Department, Federal University of Campina Grande, Campina Grande, Brazil, where he is a Professor of Electrical Engineering and Director of the Research Laboratory on Industrial Electronics and Machine Drives. In 1990, he was with COPPE, Federal University of Rio de Janeiro, and from 1990 to 1991, he was with WEMPEC, University of Wisconsin, Madison, as a Visiting Professor. His current research work is in the area of power electronics and motor drives. Dr. da Silva is currently a Member-at-Large of the Executive Board of the IEEE Industry Applications Society. He was the General Chairman of the 1984 Joint Brazilian and Latin-American Conference on Automatic Control, sponsored by the Automatic Control Brazilian Society.

Antonio Marcus Nogueira Lima (S’77–M’89) was born in Recife, Brazil, in 1958. He received the B.S. and M.S. degrees in electrical engineering from the Federal University of Paraíba, Campina Grande, Brazil, in 1982 and 1985, respectively, and the Ph.D. degree from the Institut National Polytechnique de Toulouse, Toulouse, France, in 1989. He was with the Escola Técnica Reden-torista, Campina Grande, Brazil, from 1977 to 1982, and he was a Project Engineer with Sul-América Philips, Recife, Brazil, from 1982 to 1983. From 1983 to March 2002, he was with the Electrical Engineering Department, Federal University of Paraíba. Since April 2002, he has been with the Electrical Engineering Department, Federal University of Campina Grande, Campina Grande, Paraíba, Brazil, where he is currently a Professor of Electrical Engineering. His research interests are in the fields of electrical machines and drives, power electronics, electronic instrumentation, control systems, and system identification.