Single-phase Induction Motor Drives Systems

M.B.R. Correa2, C.B. Jacobina' , A.M.N. Lima' and E.R.C. da Silva' Laboratbrio de Eletronica Industrial e Acionamento de MQquinas

Departamento de Engenharia ElCtrica, Universidade Federal da Paraiba 58109-970 Campina Grande, PB, Brasil, Caixa Postal 10105

Fax: ++55 (83)3 10- 10 15 Email: j acobina@dee.ufpb. br 2Departamento de Eletricidade, Escola TCcnica Federal de Alagoas - AL - Brasil

Abstract: This paper discusses high- and low-performance single-phase motor drbe sys- tems. The drive system configurations provide D

power factor control and improved electromag- 1

netic torque. Several high-performance control stratregies have been adapted to be used with the single-phase machine. A low-performance and low-cost strategy that does not require to remove the start-up and running capacitors of the single-

n

- - - phase machine is also proposed. Simulation and experimental results are provided to illustrate the

Fig. 1. Configuration of the single-phase drive system.

operation of the proposed drive systems.

I. INTRODUCTION

Single-phase machines are widely employed for low- power applications. In those applications the machine runs 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 (start-up and running capacitor).

The cost reduction of the semiconductor switches and the need to provide power factor control to guarantee the economic use of energy even for low-power applications has stimulated the investigation of different single-phase motor drive schemes [l-71. In those schemes the single- phase machine, without its start-up and running capacit- ors, 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 subject. Such a sys- tem will provide a high quality electromagnetic torque to the load as well as the possibility of variable frequency op- eration, which can exploited to improve efficiency of the

design of the overall drive system control strategy. The basic drive configuration being studied in this pa-

per is presented in Fig. l . This configuration is indic- ated for high-performance applications and provide power factor control and the possibility of operating in the re- generating mode. Even so, in this work it is also investig- ated another configuration (see Fig. 6) that i s indicated for low-cost low-performance applications. The use of this configuration has not yet being discussed in the literature.

Several control strategies for the configuration shown in Fig. 1 are proposed in this paper. In particular. some high-performance control strategies (like field ori- ented control and direct torque control) have been refor- mulated to be used with the single-phase machine.

Simulation and experimentally obtained results are presented to demonstrate the main characteristics of the proposed drive systems. These results validate the meth- odology and the modelling approach employed in this work.

11. MACHINE MODEL drive.

Different Static converter toPologies to SUPPlY the single-phase machine have been discussed in the literat-

tions that employ four switches. In [2] a converter is used to emulate a variable capacitor but without PO\?..,^ :actor control. In [6] switches are u s 4 to obtain a two-phase inverter but the input rectifier is not considered. This pa- per also consider the use of the topology proposed in [5,8] but the focus given here is on the detailed analysis and

The dynamic behavior of the single-phase machine is described by the following mathematical equations:

(1) ure [2,6,8]. References [2] and [6] proposed configura- d&

(2) W q

+ WbS,, (3)

v:d = r s d i z d + - dt

v:q = rsqi:q + dt d d : d 0 = r,.& + -

dt

0-7803-5160-6/99/$10.00 0 1999 IEEE. 403

(4)

$:d = 1sdi:d f m s r d i : d

d:q = Lqi:q + msrqi:q

@:d = 1ri:d + m s r d i : d

4 ~ : ~ = lri:q + msrqi:q

Te = P(i:qi:dmsrq - i:dieqmsrd)

(5)

(6)

(7)

(8)

(9)

(10) dwr P (Te - Tm) = J- + Fur . d t

Those equations were derived in the same way as the equations of the dq model for an asymmetric two-phase machine were derived [9]. The superscript s in the vari- ables of the equations (1)-(10) indicates that the stator reference frame has been employed.

111. ROTOR FLUX CONTROL

Because of the unbalance of the stator windings of the single-phase machine, the use of vector control requires some adaptation before been used with this type of ma- chine. Differently from the representation of a s~ illmetric machine the straight definition of a vector model does not exist for the case of a single-phase machine.

The result of such unbalancing can be observed in the torque waveform. Even in the case in which the currents iSsd and i:q are sinusoidal with same amplitude and r / 2 rad away from each other, an oscillatory term still exists in the electromagnetic torque. The existence of such an oscillatory term can be foreseen through equation (9) in which different values of mutual inductances are present.

To define the control strategy based on the rotor flux consider the following equation:

(11) P

T e - - -(is 1, s q 4' rd msrq - i : d d : q m s r d )

This equation has been obtained by using (7), (8) and (9). Observation of equation (11) shows that two possibil-

ities exist to eliminate the oscillatory term of the elec- tromagnetic torque. The first alternative is +? make d ( i z d ) / d ( $ q ) = m s r q / m s r d and d ( 4 i d ) / d ( 4 i q ) = 1. The second option is to make d(@d) /d(@q) = msrd/msrq and d(izd)/d(jzq) = 1. The term d ( x ) represents an op- erator that extracts the amplitude of the sinusoidal wave- form x. In both cases it is assumed that the dq compon- ents are 7r/2 rad away from each other.

To analyze the effect of these two alternatives consider the dynamic equation that relates the rotor flux to the stator currents:

where Tr = l r / rr . In equations (12) and (13) the second alternative that

requires d(i:d)/d(i:q) = 1 implies A(4:d)/d(4:q) # m,,d/msrq. Consequently the oscillatory term is not elim- inated. However, the first alternative is consistent with (12) and (13) as will be shown next.

Let iZd = Iscos(wet) and i:q = Iclssin(wet), where w e is the stator frequency and k = msrd/msrq. Now define iidl = i:d and i:ql = i : q / k . In such case the electromag- netic torque is given by

This expression is equivalent to the expression the elec- tromagnetic torque for a symmetric machine in which the oscillating term does not exists. It should be noted that in this case the rotor currents, as well as the rotor fluxes, are balanced.

Employing the current compensation as defined above, the vector model can be developed for the single-phase machine even though equations (12) and (13) are not sym- metrical. Considering that i:d = izdl and izq = k i i q l , equations (12) and (13) can be re-written as:

The vector model can be defined from equations (15) and (16). This vector model written for an arbitrary ref- erence frame (denoted by the superscript a), that is 6, radians away from the phase d of the stator, is given by:

where wa = db,/dt is the speed of the arbitrary refer- ence frame. The variables are transformed through the following expressions:

4; = d:d + j4:q = ( 4 : d + j 4 : q ) e - j 6 a

i;l = i:dl + j i:ql = ( i : d l + j i i q 1 ) e - j 6 a .

Based on the vector model given by (17), it is possible to apply the field oriented principles to control the rotor flux and the electromagnetic torque of the single-phase machine. 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 machine. In this diagram T,* a.nd 4; represent the desired electromagnetic torque and the desired amplitude of the rotor flux, re- spectively. The block e j % performs the coordinate trans- formation from the reference frame aligned along with the rotor flux vector to the stationary reference frame. Fur- thermore, izz and i;f; represent the reference currents that

404

I f.6 I

I Fig. 2. Block diagram of the indirect rotor field oriented control.

are supplied to the stator current controllers, which in turn should impose them on machine windings. The ana- lysis and design of the current controllers will be presented in section V.

IV. STATOR FLUX CONTROL

To study the stator flux control lets consider the fol- lowing expression for the electromagnetic torque:

1 1 Te = P [ Z 4 : d i : , - k 4 : q i s J d + z ( k 2 1 s q - I s d ) i z d i z q ] . (18)

This expression is obtained by using (5), (6) and (9). Due of the term i:di:q in (18), it is not possible to define a relation between 4iq and 4 : d so that the electromagnetic torque do not exhibit the oscillating term, if thc machine is supplied with sinusoidal flux waveforms. However, if & = ( b i d 1 and d ie = ( l / k ) d i q l equation (18) can be re-written as:

The coefficient (k21q - I d ) that multiplies i tdl i iql is quite small. Indeed, considering that k = m s r d / m s r q may be given approximately by the ratio of the number of turns of the respective windings, i.e., k n S d / n s q and l S d / l s q Z n,”,/nz,, then l s d Z k2 lSq . In this case, the term i id1iiql may be neglected and the expression for the torque becomes:

T e Z p [ d ~ d l i ~ g l - 4 : p l i : d i l . (20) To determine the dynamic equation that relates the

stator fluxes to the stator currents it is necessary to place (5)-(8) into (15) and (16), and to introduce dtd1 and q5iq1. The resulting equations are:

where (Tsd = 1 - m f , . d / ( l r l s d ) and U s p = 1 - ??2~, . , / (11-1sq) . As it has been done for the case of the rotor flux con-

trol, the vector model written for an arbitrary coordinate system can be derived from equations (21) and (22). The vector model for the stator flux control is given by:

The model in (23) presents an additional term C: that represents the asymmetry of the machine. Note that this term depends on ( k 2 l S q - l , d ) , which is quite small as men- tioned before. Another important remark is that the term 4; in steady-state it can be split into two vectors rotating at different frequencies. One of these vectors rotates at frequency we - w , (direct sequence) and the other rotates at frequency we + w , (inverse sequence). The vector that rotates at frequency w e - w, can be easily compensated if the stator flux controller is implemented in the syn- chronous reference frame and has an integrating term. However, the vector that rotates at frequency we + w , cannot be compensated in this way. Though, the coeffi- cient (k21,, - l s d ) that multiplies the term which rotates a t frequency we t w , is quite small and should not disturb significantly the stator flux control loop.

A . Direct field oriented control Based on the vector model given by equation (23) it

is possible to apply the field oriented principles to con- trol the stator flux of the single-phase machine. Fig. 3 shows the block diagram of the direct stator field oriented control scheme that has been adapted for the single-phase machine. In this diagram T,* and 4:1 represent the desired electromagnetic torque and the desired amplitude of the stator flux, respectively. The amplitude of the stator flux is given by q5sl = Jm and the angle of the reference frame is determined by 6,j = tan-1(4iql/#dl). This angle is employed in the block e j 6 a f to perform the coordinate transformation from the reference frame aligned along with the stator flux vector to the station- ary reference frame. The stator flux components are de- termined by integrating & - rSiid and vZq - rsizq. The performance of the control scheme sketched in Fig. 3 can be improved if a compensation for the disturbing term in (23) is provided. Just as mentioned in the case of the rotor flux control, i;; and i i i represent the reference currents supplied to the stator current controllers.

B. Direct torque control The adaptation of the direct torque control strategy for

the single-phase machine is illustrated by the block dia-

405

Fig. 3. Block diagram of the direct stator field oriented control.

Fig. 4. Block diagram of the direct torque control.

gram shown in Fig. 4, in which the stator flux loop is closed around the compensated flux components f$:d1 and q5:ql. In this case the dynamic equations that relate the stator flux components to the stator voltages are given by (1) and (2) while the electromagnetic torque is determ- ined by (20). The controllers indicated in Fig. 4 can be implemented as standard PI controllers or with analog circuitry using hysteresis approach.

C. Volts/Hertz control A control strategy as simple as the Volts/Hertz scheme

can also be adapted to be applied the single-phase ma- chine. In this case, it is required to compensate the amp- litude of vi: and vi; by the factor k = m s r d / m s r q , that is

sd - - (W:4zl) COS(W:t), = (wZd:l/k) sin(w:t). (25)

The compensation of the amplitude of vi; will be dis- cussed in next section.

V. STATOR CURRENT CONTROL

The stator current control loop is required for the con- trol strategies illustrated in Figs. 2 and 3. The stator current controller can be implemented by using either an

analog hysteresis based controller or a discrete-time con- troller [7]. To design a stator current controller it is ne- cessary 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 cur- rent controllers is derived from equations (1) to (8) and is given by:

L q ( 1 - usq) di” = ( r sq + )izq + usqlsq 2 + e : q . (27)

Tr

The counter electro-motive forces ezd and e:q are given by

Considering that 4 : d l = # d and 4:ql = k4:q then, from (1) and (2) it can be seen that v : d l = and viql = kv:,, respectively, which can be used as an approximate relationship for compensating the stator voltage compon- ents. This voltage compensation minimizes the oscillating term of the eletromagnetic torque. Similarly, new com- pensated components of the counter electro-motive forces can be defined as e:dl = ezd and e;ql = ke;,. Introducing the compensated voltage and electro-motive forces com- ponents in (26) and (27), the following vector model can be written

where

Notice that, except in low-speed range, the term rep- resented by is small since it depends on (k21sq - l s d ) .

In terms of the vector model given by (30), the machine asymmetry is represented by the term dz. In the same way that was discussed for C:, this term, in steady state, is constituted by two vector of frequencies we - wa (direct sequence) and we + w, (inverse sequence).

When the machine is symmetrical and a proportional- integral controller is employed the use of the synchronous reference frame has proved to be the best choice just be- cause the disturbance terms are transformed to dc quant- ities that are easily compensated by t.he controller itself.

406

In the case of the single-phase machine the use the syn- chronous reference frame (w, = w e ) only solves the dis- turbance rejection for the direct sequence term that ro- tates at the frequency w, - w, and after coordinate trans- formation becomes a dc component. The inverse seq.uence term, after coordinate transformation becomes a compon- ent that rotates at 2w, and consequently cannot be com- pensated by a single controller.

The previous analysis has shown that in the case of the single-phase machine it can be necessary to invest- igate new controller structures which provide better dis- turbance rejection properties. The controller structure proposed in this paper employs two different synchronous controllers. The positive sequence synchronous controller rotates at +we and is designed to compensate the dir- ect sequence term. The negative sequence synchronous controller rotates at -we and is designed to compensate the inverse sequence term. These two controllers operate simultaneously and its outputs are added.

Fig. 5 shows the block diagram of the stator current control loop. The input reference currents i:$:, i::: and sdl, iJ"i: indicated in this diagram are obtained at the

outputs of block diagrams of Figs. 2 and 3 by using the coordinate transformation e- j6$ and e-j6T , respectively. The superscripts + and - of the variables of Fig. 5 indic- ates that the positive and negative synchronous reference frame, respectively, has been adopted.

The outputs of the current controller provide the mod- ulating waveforms for the pulse width modulator that in this case is implemented with the technique proposed in [7].

ie-"

VI. LOW-COST LOW-PERFORMANCE DRIVE SCHEME

The control strategies discussed in the previous sec- tion were proposed for high-performance single-phase mo- tor drive system in which its capacitors are removed. However, in some applications the performance needed does not require a large step towards a high performance single-phase drive scheme. The configuration presented in Fig. 6 is conceived for such mid-range applications. In this configuration the single-phase machine can be sup- plied with variable frequency and and amplitude voltages to permit a more effective speed change.

This configuration has the following advantages with re- spect to the standard single-phase operating conditions: i) It provides power factor control and ii) It may supply the windings with variable frequency and amplitude voltages.

However, it should be noticed that this scheme do not permit to supply the machine with independent phase voltage. Thus, only a standard Volts/Hertz control, i.e., with same amplitude voltage, must be considered.

VII. SIMULATION RESULTS

Due the space limitations, only few simulations res- ults for the high-performance configuration (Fig. 1) are

-& +

Fig. 5. Block diagram of the current controller. - I

--I---- I Fig. 6. Configuration of the low-cost low-performance single-phase drive system.

presented. This results were obtained with the machine Machine I , that have k = 0.75. Figs. 7, 8 and 9 show the transient waveforms observed when the Machine I is star- ted from standstill and is controlled with the strategies of Figs. 2, 3 and 4, respectively. The current controller used in the strategies of Figs. 2 and 3 was a hysteresis- type controller. The flux controller used in Fig. 3 was a continuous-time PI controller. Both controllers in Fig. 4 were of hysteresis-type. Both the rotor flux reference (Figs. 7) and the stator flux reference (Figs. 8 and 9) are constant and equal to 0.4Wb. The reference for the elec- tromagnetic torque is given by T,* = 4N.m for t 5 0.4s and T,* = 6N.m for t > 0.4s. The slower response for the rotor flux and electromagnetic torque at the begin of the transient is a characteristic of the indirect field oriented control, when the machine starts without proper magnetic excitation. These results show that it is possible to apply these strategies to the control of a single-phase motor.

$ v

z 0 . 2 O m 3 t /

Fig. 7. Torque and rotor flux waveforms (Scheme Fig. 2).

"0 0 . 2 0 . 4 0.6 0 . 8

20.3

z0 .2

-

2 0; 0.2 0 . 4 0.6 0 . 8 t i m e Is)

Fig. 8. Torque and stator flux waveforms (Scheme Fig. 3).

The current control strategy that employs simultan- eously two synchronous controllers (composed control- ler, Fig. 5) has been tested by simulation (ideal voltage source) and the results were satisfactory. Figs. 9 shows the comparaison of this strategy with the strategy that use a single synchronous controller for step changes (t = Os and t = 2s) of the reference currents. Figs. 10 shows the waveforms of i:: - i:d for the composed and the single controllers. As it can be noticed, the disturbance rejection was significantly improved (current error < 0.04%) with respect to the strategy that utilizes a single synchronous controller. However, it must be remarked that the use of a single synchronous controllers provides a very acceptable reduction of the disturbance terms (current error E 0.8%).

VIII. EXPERIMENTAL RESULTS

In addition to Machine I another machine w&s em- ployed in the experimental tests. Machine I1 (k = 0.90). Fig. 11 presents the stator current of Machine I : i) The machine being supplied directly from the single-phase grid (Fig. l l a ) ; ii) Configuration of Fig. 6 at 60H.z (Fig. l lb ) ;

h I 0.2 0 . 4 0.6 0 . 8

20.3

z 0 . 2

U 20.1

t i m e (s)

Fig. 9. Torque and stator flux waveforms (Scheme Fig. 4).

4 ; 0.1 $ 0

2-0.2 v) -0.1 '1

I

a 0 1 2 3 4 i -

Fig. 10. Current error. Single synchronous controller (top). Com- posed controller (bottom).

and iii) Configuration of Fig. 6 at 40Ht (Fig. l lc) . We observe that the currents are distorted in all the cases. With the configuration of Fig. 6 it was be possible to re- duce the speed of the machine to 40Ht by reducing the stator frequency.

Figs. 12 to 13 present izd, i:q and their respective errors

controller for the Machine I I . The reference currents are given by ii: = 2.75cos(60st), iz; = 2.50sin(60nt), 0 5 t < tmax/2 and i:: = 1.37cos(60nt), iz; = 1.25sin(60nt), tmax/2 5 t 5 tmax where the currents are given in amperes and t,,, is the total time of the experimental test.

It can be observed that the controllers present max- imum error of 4%. The results are considered to be satis- factory.

iS s d * - i : d , i;; - i$ obtained with a single synchronous

IX. CONCLUSION

This paper has discussed several control strategies for single-phase motor drive systems. The modelling ap- proach that has been proposed made possible do adapt

408

8

4

0 (4

-4

-8 0 0. U2 0.04 0.06

I 0.02 0.04 0.06

8f I

-8 I 0 0.02 0.04 0.06

time (s)

Fig. 11. Machine currents. (a) Single-phase grid (60Hz). (b) Low- cost scheme (60Hz). (c) Low-cost scheme (40Hz).

0 . 1 0.2 0.3 0 . 4 0 . 5 time Is)

Fig. 12. Measured current and error for d-axis.

time (s)

Fig. 13. Measured current and error for q-axis.

some high-performance control strategies to be used with a single-phase motor drive system. This approach also provided a simple representation of the single-phase ma- chine asymmetry that was very useful to understand some features of a single-phase drive system and to propose a stator current control strategy that has improved disturb- ance rejection properties. It is important to remark that the same approach can be used to represent the asym- metry of a three-phase machine. The paper also proposed a low-cost low-performance configuration. All the theor- etical results have been demonstrated through simulation but due to space limitations only few results were presen- ted. The experimental tests were considered satisfactory and have confirmed the claimed features.

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