Decoupling Vector Control of Single-Phase Induction Motor Drives

S. Vaez-Zadeh University of Tehran

Department of Electrical and Computer Engineering, Tehran – IRAN, Fax: +98 21 8778690

Email: vaezs@ut.ac.ir

Sh. Reicy Harooni

University of Tehran Department of Electrical and Computer Engineering,

Tehran – IRAN, Fax: +98 21 8778690 Email: shreicy@ece.ut.ac.

Abstract—By the advent of power electronic and microelectronic devices and circuits, together with their steadily cost reduction, single phase induction motors (SPIMs) increasingly are considered as variable speed drives in recent years. This provides improved performances and energy efficiency in traditional applications and opens the door for new applications. Vector control proved to be practical and effective in SPIM drives. However, winding asymmetry in SPIMs causes extra coupling between two stator windings and results in unbalanced machine operation and high current and torque pulsations. In this paper an indirect vector control system is proposed for SPIMs including a rather simple and effective decoupling scheme. This is achieved by introducing two new decoupling signals to the system in addition to the decoupling signals similar to ones used in three phase induction motor drives. The proposed decoupling vector control system is applied to a 0.5 hp commercial SPIM and its performance is evaluated by extensive simulation. It is shown that the proposed decoupling scheme substantially improves the motor performance by providing balanced operation and smooth currents and torque.

Keywords—Single-phase induction motors, motor drives, winding asymmetry, decoupling system, vector control.

I. INTRODUCTION SINGLE-PHASE induction motors (SPIMs) have been widely used in low-power applications for many years. In these applications the machines operate at fixed speed and low efficiency and consume about 10% of electrical energy used in all kinds of motors. It is estimated that 1.7 billion KWh/year of energy is saved if the efficiency of these motors increases by only one percent [1]. By the advent of power electronic and microelectronic devices and circuits, together with their steadily cost reduction and future energy challenge, a number of efficient SPIM drives have been introduced in recent years [2-9].

A common configuration for such drives is shown in Fig. 1 which uses a conventional inverter to supply the machine main and auxiliary windings. This provides improved performances and energy saving in traditional applications and opens the door for new applications [10].

Vector control proved to be practical and effective in SPIM drives [11]. However, winding asymmetry in SPIMs causes extra coupling between two stator windings and results in unbalanced machine operation. This, in turn, prod-

110/220 VAC60/50 Hz

Main Aux.

C

Fig. 1. Configuration of single-phase induction motor drive.

uces current and torque pulsations and limits SPIM drive applications [12]. It is suggested to use hysteresis current control to overcome the problem [13]. However, system performance under hysteresis control would not be satisfactory in a light load condition which is the case most of the time. A current double-sequence control is also proposed to eliminate the pulsations [14]. However, it is rather complex due to additional controllers and extensive on-line computation.

In this paper an indirect vector control system is proposed for SPIMs including a rather simple and effective decoupling scheme. This is achieved by introducing two decoupling signals to the system. The proposed decoupling vector control system is applied to a 0.5 hp commercial SPIM and its performance is evaluated by extensive simulation. It is shown that the proposed decoupling scheme substantially improves the motor performance.

II. MACHINE MODEL The dynamic model of single phase induction machines

in a stationary reference frame can be represented as [15]:

00

s s sasas as as

s s sbsbs bs bs

Rv ip

Rv iλλ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(1)

0000

s sr ar ar

s sr br br

R ip

R iλλ

⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤= +⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

0 11 0

sar

r sbr

λω

λ⎡ ⎤⎡ ⎤

+ ⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦ (2)

7330-7803-9033-4/05/$20.00 ©2005 IEEE.

0 00 0

s s sas asras as ar

s s sbs bsrbs bs br

L Li iL Li i

λλ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤

= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(3)

0 00 0

s s sasr rar as ar

s s sbsr rbr bs br

L Li iL Li i

λλ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤

= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(4)

( )2s s s s

e bsr bs ar asr as brPT L i i L i i= − (5)

( )2 e L r rP T T Jp Bω ω− = + (6)

Variables sasv , s

bsv , sasi , s

bsi , sari , s

bri , sasλ , s

bsλ , sarλ , and

sbrλ are auxiliary and main voltages, currents and fluxes of

the stator and rotor in the stationary reference frame respectively. asR , bsR and rR denote the stator and rotor resistances; asL , bsL , asrL and bsrL represent the stator and rotor self and mutual inductances; rω , eT and LT are the rotor speed, the electromagnetic torque and the load torque; and P , J and B are the number of machine poles, the moment of inertia and the viscous friction coefficient respectively. Also p is a derivation operator.

It is seen that there is model asymmetry due to the unequal resistances and inductances of the main and auxiliary windings. A part of the model asymmetry can be eliminated by referring all variables to the auxiliary winding. The new mathematical model employing such transformation is obtained as:

'' ' '

00

s s sasas as as

s s sbsbs bs bs

Rv ip

Rv iλλ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(7)

' ''

' ''

0 00 0

s sar arrs s

br brr

iRp

iRλλ

⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤= +⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

'

'

0 11 0

sar

r sbr

λω

λ⎡ ⎤⎡ ⎤

+ ⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦ (8)

'

'' ' '

0 00 0

s s sas masas as ar

s s sbs masbs bs br

L Li iL Li i

λλ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤

= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(9)

' ''

' ' ''

0 00 0

s s smasar as arr

s s smasbr bs brr

L i iLL i iL

λλ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤

= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(10)

( )' ' '

2s s s s

e mas bs ar as brPT L i i i i= − (11)

where:

' ' ' , , s s s s s sas as bsbs bs bs bs bs bs

bs bs as

N N Nv v i i

N N Nλ λ= = =

' ' ' , , s s s s s sas as rar ar ar ar ar ar

r r as

N N Nv v i i

N N Nλ λ= = =

' ' ' , , s s s s s sas as rbr br br br br br

r r as

N N Nv v i i

N N Nλ λ= = = (12)

2 2

' ' , as asbs bs r r

bs r

N NR R R R

N N⎛ ⎞ ⎛ ⎞

= =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

2 2

' ' , as asbs bs r r

bs r

N NL L L L

N N⎛ ⎞ ⎛ ⎞

= =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

(13)

In the above equations masL denotes the auxiliary winding magnetizing inductance. asN , bsN and rN represent the number of stator and rotor windings respectively. It can be shown that '

as bsL L≅ [14]. Thus, the asymmetry in the inductance matrices in (9) will be eliminated.

Referring to (11), if the following equation is held between the currents of main and auxiliary windings:

90s sbsas bs

as

Ni i

N= ∠± (14)

then the motor works under a balanced condition with no torque pulsation. In fact, the backward rotating magnetic field, as the cause of the torque pulsation and increased copper loss vanishes. Therefore, the implementation of the vector control in SPIMs becomes a practical task.

The vector control of SPIMs needs a machine model in a rotating reference frame. The dynamic model of single phase induction machines in a synchronously rotating rotor flux reference frame can be express as:

e e eqqs qdsqs qs qs

e e edqs ddsds ds ds

R Rv ip

R Rv iλλ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

0 11 0

eqs

e eds

λω

λ⎡ ⎤⎡ ⎤

+ ⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦ (15)

' ''

' ''

0 00 0

e eqr qrre e

dr drr

iRp

iRλλ

⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤= +⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

( )'

'

0 11 0

eqr

e r edr

λω ω

λ⎡ ⎤⎡ ⎤

+ − ⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦ (16)

'

'

0 00 0

e e es msqs qs qr

e e es msds ds dr

L Li iL Li i

λλ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤

= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(17)

' ''

' ''

0 00 0

e e emsqr qs qrr

e e emsdr ds drr

L i iLL i iL

λλ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤

= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(18)

( )' '

2e e e e

e ms qs dr ds qrPT L i i i i= − (19)

where Ls = Las and Lms = Lmas. A superscript e shows that the variables are referred to a rotary reference frame with an angular velocity of ωe. The equations are similar to the

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equations of a three phase induction machine except that the stator voltage equations contain time dependent resistance terms, Rqqs, Rqds, Rdqs and Rdds due to unequal resistances of stator main and auxiliary windings. These terms can be presented in terms of stator windings resistances as:

( )' '

cos 22 2

as bs as bsqqs e

R R R RR θ

+ −= − (20)

( )' '

cos 22 2

as bs as bsdds e

R R R RR θ

+ −= + (21)

( )'

sin 22

as bsdqs qds e

R RR R θ

−= = (22)

If the referred resistance of main winding is equal to the resistance of auxiliary winding, the time dependent terms vanish. Also, if the magnitude of the variable terms in (20)-(22) are small with respect to the constant terms, i.e.:

' '

2 2as bs as bsR R R R− +

<< (23)

it may be possible to neglect the difference of stator windings. However, (23) is not held for most SPIMs. Therefore, time dependent terms must be compensated in vector control of a SPIM to achieve a smooth machine performance as will be presented later on.

III. VECTOR CONTROL A vector control system in rotor flux reference frame is

described by introducing rotor flux linkage components as:

' ' ' 'r r rqr ms qs r qrL i L iλ = + (24) ' ' 'r r rdr ms ds r drL i L iλ = + (25)

The motor torque in terms of the flux linkage components is then obtained as:

( )' ' ''2

r r r rmse dr qs qr ds

r

LPT i iL

λ λ= − (26)

The d-axis of the reference frame is oriented along the rotor flux vector i.e.:

' ' & 0r rdr r qrλ λ λ= = (27)

Thus, the torque equation of (19) is reduced to:

'2rms

e r qsr

LPT iL

λ= (28)

and the rotor voltage equations become:

( )' ' ' ' 0r r rqr r qr e r drv r i ω ω λ= + − = (29) ' ' ' ' 0r r rdr r dr drv r i pλ= + = (30)

Finally from (13), (14), (16), (18) and (19) the rotor flux linkage and slip speed are obtained as:

'

1

rr ms ds

drr

L ip

λτ

=+

(31)

'

rms qs

sl rr dr

L iω

τ λ= (32)

The block diagram of indirect vector control system of SPIMs is shown in Fig. 2 where the decoupling signals

Dcpldsv and Dcpl

qsv are described in the next section. It is seen that

the two current controllers provide control voltages Ctrldsv and

Ctrlqsv . These voltages are supplemented by decoupling

voltages Dcpldsv and Dcpl

qsv respectively to produce voltage commands in the rotating reference frame. The latter voltages are then transformed to the stationary reference frame of the auxiliary windings. Finally, the main winding voltage is referred to its own winding.

IV. DECOUPLING SYSTEM In vector control of SPIMs, like three phase motors, the

stator currents must be controlled. If the employed inverter is of the voltage type with hystersis current controllers, it is possible to use the current command signals to obtain the inverter gating signals. However, if the inverter is of PWM type with voltage control, the voltage command signals should be produced as the outputs of the current controllers. These voltages contain variable terms as seen in (15)-(22). These terms, which are not exist in three phase induction motors, deteriorate the desired functioning of the current controllers and lead to a lack of accuracy in current control. Therefore, a decoupling system is proposed in this section to overcome the problem.

*

*eT

iqsCalculation

Calculationsl

sl

r

*dr

dr

qsi *

e

dq

ab *vbs

asv*

*ids

dsi

ds*vds

qsv*

iqs

n

v

L

dr

qsi

r

ms

r

ms

Te

dr

2 L *

PL

Ctrl

Dcplvds

Ctrlvqs

Dcplvqs

'

'

'

'

Fig. 2. Vector control system of SPIM.

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Stator voltages can be rearranged by eliminating the rotor current and stator flux linkages using (15) in connection with (17) and (18), thus:

( ) ( )( )' '-

- cos 2 sin 22 2

r r r ras bs as bsqs qs qs e ds e

R R R Rv i i iθ θ

+= + +

' '' '

r r r rms mss qs qr e s ds dr

r r

L Lp L i L i

L Lσ λ ω σ λ⎛ ⎞ ⎛ ⎞

+ + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(33)

( ) ( )( )' '

sin 2 cos 22 2

r r r ras bs as bsds ds qs e ds e

R R R Rv i i iθ θ

+ −= + +

' ' ''

r r r rms mss ds dr e s qs qr

r r

L Lp L i L i

L Lσ λ ω σ λ

⎛ ⎞⎛ ⎞+ + − +⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠ (34)

The above equations can be rearranged as:

'

2r r Sym Asymas bsqs s qs qs qs

R Rv L p i v vσ

⎛ ⎞+= + + +⎜ ⎟⎝ ⎠

(35)

'

2r r Sym Asymas bsds s ds ds ds

R Rv L p i v vσ

⎛ ⎞+= + + +⎜ ⎟⎝ ⎠

(36)

where

Sym r r rms msqs qr e s ds dr

r r

L Lv p L i

L Lλ ω σ λ

⎛ ⎞= + +⎜ ⎟

⎝ ⎠ (37)

Sym r r rms msds dr e s qs qr

r r

L Lv p L i

L Lλ ω σ λ

⎛ ⎞= − +⎜ ⎟

⎝ ⎠ (38)

( ) ( )( )'

cos 2 sin 22

Asym r ras bsqs qs e ds e

R Rv i iθ θ

−= − + (39)

( ) ( )( )'

sin 2 cos 22

Asym r ras bsds qs e ds e

R Rv i iθ θ

−= + (40)

It is seen that (37) and (38) are similar to the decoupling signals in vector control of three phase induction machines [16]. However, (39) and (40) appear specifically in SPIMs model due to windings asymmetry. They are time varying voltages with a frequency double the inverter output frequency. These voltages vanish only if '

as bsR R= as in a two phase induction machine. Therefore, the decoupling signals for SPIMs are determined as following:

Dcpl Sym Asymds ds dsv v v= + (41) Dcpl Sym Asymqs qs qsv v v= + (42)

A synthesis of the decoupling signals based on (26)-(30) is shown in Fig. 3.

e

qr

i qs

r

ms

LL

msL

-Cos e2 Sin e2

Sin e2 Cos e2

asR bs-R'

2

dr

r

ms

LL

idsmsL

qsv Dcpl

qsv Asym

dsv Dcpl

dsv Asym

p

p

Fig. 3. Synthesis of decoupling signals.

V. PERFORMANCE EVALUATION The proposed decoupling vector control system is applied

to a commercial 0.5 hp single phase induction motor with parameters shown in Table I. Extensive simulation results are presented to evaluate the motor drive performance.

Decoupling voltages are shown in Fig. 4. It is seen that the voltages are pulsating due to their asymmetric components in accordance to (39) and (40). A trajectory of asymmetric voltage components are shown separately in Fig. 5. It is seen that the magnitude of d- and q- axis components are the same at about 0.06 p.u.. This value is small in comparison with winding voltages vas and vbs shown in Fig. 6. However, these small voltages have substantial effects on machine performance.

Fig. 7 shows d- and q- axis stator current components with and without asymmetric voltages included in the decoupling signals. It is seen that in the absent of asymmetric voltages, both current components contain serious pulsations; whereas the pulsations die out when asymmetric voltages are included in the decoupling signals. The balancing effect of the asymmetric voltages can also be seen in the trajectory of stator currents presented in Fig. 8. The figure shows that the current trajectory forms a circle, instead of an ellipse, when asymmetric signals are used. Such balancing effect improves the motor electromagnetic torque, as shown in Fig. 9, where the torque pulsations are eliminated. The harmonic content of motor torque with and without asymmetric voltage signals can be seen in Fig. 10. It is evident that a much smoother torque is developed as a result of applying asymmetric signals. The motor speed is shown in Fig. 11. The current of stator windings are also shown in Fig. 12 under the balanced operation. It is seen that the currents are 90° out of phase and the ratio of their amplitude is equal to the winding turn ratio.

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VI. CONCLUSION Winding asymmetry in a single phase induction machine

causes unbalanced machine operation and current and torque pulsations, even under a vector control system. The problem can be overcome by the proposed decoupling system producing extra signals in addition to decoupling signals used in three phase induction motor drives.

ACKNOWLEDGMENT The support of Center of Excellence on Applied Electromagnetic Systems at the university of Tehran is appreciated.

TABLE I. SINGLE-PHASE INDUCTION MACHINE PARAMETERS.

1425 r.p.m. nN 370 Watt nP

5.3 A nI 220 V nV

2.48 N.m. nT 50 Hz nf

14.75 Ω bsR 5.2 Ω asR

4.12 Ω 'brR 7.5 Ω '

arR

11.8 mH lbsL 17.9 mH lasL

6.68 mH 'lbrL 11.8 mH '

larL

168 mH mbsL 300 mH masL

0 N.m.s/rad B 0.02488 kgm2 J

0.749 asbs

NN 4 P

0 0.1 0.2 0.3 0.4-0.25

0.25

0.75

Time (sec.)

V sDcc

l (p.u

.)

vdsDcpl v

qsDcpl

Fig. 4. Decoupling voltages.

-0.1 -0.05 0 0.05 0.1-0.1

-0.05

0

0.05

0.1

vdsAsym (p.u.)

v qsAsy

m (p

.u.)

Fig. 5. Trajectory of asymmetric components of decoupling voltages.

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

vas (p.u.)

v bs (p

.u.)

Witout asym. terms

With asym. terms

Fig. 6. Trajectory of stator voltages.

0

1.5

3

i ds (p

.u.)

0 0.1 0.2 0.3 0.40

1.5

3

Time (sec.)

Without asym. terms

With asym. terms

(a)

-1

1

3

i qs (p

.u.)

0 0.1 0.2 0.3 0.4-1

1

3

Time (sec.)

Without asym. terms

With asym. terms

(b)

Fig. 7. Stator current components; a) d- axis current, b) q- axis current.

-2 -1 0 1 2-2

-1

0

1

2

ias (p.u.)

i bs (p

.u.)

Without asym. termsWith asym. terms

Fig. 8. Trajectory of stator currents.

737

-1

1

3

Te (p

.u.)

0 0.1 0.2 0.3 0.4-1

1

3

Time (sec.)

With asym. terms

Without asym. terms

Fig. 9. Electromagnetic torque.

0

10

20

30

0.01 0.1 1 100

10

20

30

fn (kHz)

| Te,

n | / T

Base

(%)

Without asym. terms

With asym. term

Fig. 10. Harmonic content of motor torque.

0 0.1 0.2 0.3 0.40

0.5

1

1.5

T ime (sec.)

ωr (p

.u.)

Act. speedRef. speed

Fig. 11. Motor speed.

-1

0

1

i s (p.u

.)

0 0.1 0.2 0.3 0.4-1

0

1

Time (sec.)

ibs

ias

Fig. 12. Stator currents.

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[14] M. B. Correa, C. B. Jacobina, E. R. C. daSilva, A. M. N. Lima, " Vector Control Strategies for Single-Phase Induction Motor Drive Systems," IEEE Transactions on Industrial Electronics, vol. 51, no. 5, Oct. 2004, pp.1073-1080

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