608 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. … Shoubra/Electrical Engineering/916...Open-loop speed control of the two-phase IM using two single-phase half-bridge inverters operated

608 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 24, NO. 3, SEPTEMBER 2009

An Unsymmetrical Two-Phase Induction MotorDrive With Slip-Frequency Control

Naser M. B. Abdel-Rahim and Adel Shaltout

Abstract—This paper proposes a closed-loop control strategyto operate an off-the-shelf single-phase induction motor (IM) as asymmetrical two-phase IM. The proposed control strategy employsthe SFC technique to independently control the stator currents ofboth the main and auxiliary windings, and make them follow a pre-defined sinusoidal waveform. Simulation and experimental resultsshow that the proposed scheme is successful in operating the con-ventional single-phase IM as a symmetrical two-phase IM with fastdynamic and transient responses. In addition, the proposed controlsystem achieves cost-effectiveness in both initial and running costs.

Index Terms—Electric motor drive, single-phase induction mo-tor (IM), slip-frequency control (SFC), unsymmetrical two-phaseIM.

I. INTRODUCTION

NUMEROUS investigations have been carried out in theliterature to combine the merits of the polyphase induc-

tion motors (IMs) (i.e., high performance) with those of thesingle-phase IMs (i.e., widespread use and availability) [1]–[7].These investigations have offered several control strategies andcircuit topologies to operate the single-phase IM as a two-phasemotor. The resulting two-phase IM drive has achieved cost-effectiveness in both initial and running costs. Initial costs havebeen reduced by: 1) dispensing the need to manufacture espe-cially designed two-phase symmetrical motors and 2) increas-ing the output power of the single-phase motor (see Fig. 1),thus allowing the use of a lower frame size to drive the sameload. Running costs have been reduced by reducing the motorlosses and improving its power factor, thus enhancing the motorefficiency [8].

Open-loop speed control of the two-phase IM using twosingle-phase half-bridge inverters operated in the square-wavemode has been reported in [1] and [2]. With a square-wave volt-age applied at the motor terminals, the motor terminal voltageand hence, line current had high harmonic content. This resultedin increased torque harmonics as well as reduced motor overallefficiency. To alleviate such problem, a phase difference angle(PDA) control employing the pulsewidth modulation (PWM) se-lective harmonic elimination technique has been reported in [3]

Manuscript received September 6, 2006; revised January 8, 2009. Currentversion published August 21, 2009. The work of N. M. B. Abdel-Rahim wassupported by the United Arab Emirates University under Research Grant 04-04-7-11/04. Paper no. TEC-00416-2006.

N. M. B. Abdel-Rahim is with the Department of Electrical Engineering,Faculty of Engineering at Shoubra, Cairo, 11240, Benha University, Egypt(e-mail: [email protected]).

A. Shaltout is with the Department of Electrical Power and Machines, CairoUniversity, Giza 12613, Egypt (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEC.2009.2026599

Fig. 1. Torque–speed characteristics are improved when the single-phase IMis operated as an unsymmetrical two-phase motor [8].

and [4], where the motor torque has been controlled by vary-ing the PDA of the motor terminal voltages. This scheme hassuffered from degraded efficiency when the PDA between theterminal voltages is small.

Although the PDA control scheme has been successful inoperating the two-phase motor with a variable frequency, it iscomplex to implement. Moreover, high-torque pulsations re-sult when the phase angle between the motor terminal voltagesis other than 90. Torque pulsations become even more pro-nounced when the phase angle is small.

Space-vector closed-loop speed control of symmetrical two-phase IMs has been reported in [5]. Though successful in con-trolling the speed of the motor, this control scheme has not beenoptimal in driving the unsymmetrical two-phase IM since it ne-glects the zero-sequence components of the voltages/currents.Taking the zero-sequence components into account has com-plicated the implementation of the digital current controller [5].Rotor-flux-oriented space-vector control of unsymmetrical two-phase IMs has been reported in [6] and [7]. Although this schemeproduces a high-dynamic-performance drive, yet it is compu-tationally intense. Consequently, it requires high-power micro-controller or microprocessor for its implementation.

In this paper, the slip-frequency control (SFC) technique isproposed to independently control the stator currents of boththe main and auxiliary windings, and make them follow a pre-determined sinusoidal wave. By maintaining certain operatingconditions, the proposed control scheme reduces the inherenttorque oscillations of the two-phase unsymmetrical IM, thusmaking it behave like its symmetrical counterpart. Simulationand experimental results show that this control scheme pro-vides excellent speed regulation with fast transient and dynamic

0885-8969/$26.00 © 2009 IEEE

ABDEL-RAHIM AND SHALTOUT: AN UNSYMMETRICAL TWO-PHASE INDUCTION MOTOR DRIVE WITH SLIP-FREQUENCY CONTROL 609

Fig. 2. Cross section of the unsymmetrical two-phase motor.

responses. In addition, this control strategy limits the motorline current during transient and dynamic stages to a maxi-mum of twice the full-load current, which reduces the ratingof the power inverter, and hence, its cost. To the best of theauthors’ knowledge, speed control of an unsymmetrical two-phase IM using the SFC scheme has not been reported in theliterature.

II. BALANCED OPERATION OF THE UNBALANCED

TWO-PHASE MOTOR

The objective of this paper is to use the SFC scheme to re-alize a high-performance two-phase IM drive, which makes theunsymmetrical (unbalanced) two-phase IM behave like its sym-metrical (balanced) counterpart. To meet this objective, certainoperating conditions have to be satisfied first.

Fig. 2 shows a cross section of the two-phase unsymmetricalmotor (conventional single-phase IM). The stator windings areunsymmetrical, but they are orthogonal with 90 electrical de-grees apart. The stator MMF along a position defined by angleθ (where θ = 0 defines the axis of the main winding) is givenby

F (θ, t) = F1(θ, t) + F2(θ, t)

= i1N1 cos θ + i2N2 cos(θ + 90o) (1)

where F1(θ, t) is the MMF produced by the main winding,F2(θ, t) is the MMF produced by the auxiliary winding, i1and i2 are the currents in the main and auxiliary windings,respectively, and N1 and N2 are the effective numbers of turnsof the main and auxiliary windings, respectively.

Equation (1) shows that the MMFs produced by the main andauxiliary windings have different magnitudes since the statorwindings have different numbers of turns and parameters values.These unequal MMFs result in an elliptical MMF in the motorair gap, and hence, produce inherent torque pulsations.

For the main and auxiliary currents given by

i1 = I1 max cos(ωt) =√

2I1 cos(ωt) and

i2 = I2 max cos(ωt − γ) =√

2I2 cos(ωt − γ), (2)

Fig. 3. Revolving fields in the unsymmetrical two-phase motor.

(1) can be rewritten as

F (θ, t) =1√2

(N1I1 + N2I2 sin γ) cos(ωt + θ)

−N2I2 cos γ sin(ωt + θ)

+

(N1I1 − N2I2 sin γ) cos(ωt − θ)

+N2I2 cos γ sin(ωt − θ)

(3)where the terms with cos(ωt + θ) and sin(ωt + θ) are the forwardrotating fields and the terms with cos(ωt− θ) and sin(ωt− θ) arethe backward rotating fields, and γ is the phase angle betweenthe current in the main winding and the current in the auxiliarywinding.

In order to produce a circular rotating field in the motor airgap, and hence, eliminate the torque pulsations, the backwardcomponents of the stator MMF should be canceled. This can beachieved if the following relationships between the current inthe main winding and the current in the auxiliary winding aremaintained:

I1N1 = N2I2 or I2=a × I1 (4)

and

γ = 90o (5)

a =N1

N2. (6)

Substituting (4) and (5) in (3) gives

Ff (θ, t) =√

2N1I1 cos(ωt + θ). (7)

Equation (7) shows that the resultant motor MMF containsonly the forward revolving component Ff (θ, t).

III. EQUIVALENT CIRCUIT OF THE TWO-PHASE

UNSYMMETRICAL IM UNDER SYMMETRICAL

OPERATING CONDITIONS

With both the auxiliary and main windings excited, the pul-sating field produced by each winding of the unsymmetricaltwo-phase IM can be resolved into a forward and a backwardrevolving field. Consequently, there are four revolving fields inthe air gap of the unsymmetrical two-phase IM, as shown inFig. 3.


Fig. 4. Equivalent circuit of the unsymmetrical two-phase IM [9]–[11].

Fig. 4 shows the equivalent circuit of the unsymmetrical two-phase IM with both windings excited [9]–[11], where−−→

Ef 1 induced electromotive force (EMF) in the forwardbranch of the main winding by its forward revolvingfield;−−→

Es1 induced EMF in the forward branch of the mainwinding by the forward revolving field of the aux-iliary winding;−−→

Es2 induced EMF in the backward branch of the mainwinding by the backward revolving field of the aux-iliary winding;−→

Eb1 induced EMF in the backward branch of the mainwinding by its backward revolving field;−−→

Ef 2 induced EMF in the forward branch of theauxiliary winding by its forward revolvingfield;−−→

Es3 induced EMF in the forward branch of the auxiliarywinding by the forward revolving field of the mainwinding;−→

Eb2 induced EMF in the backward branch of theauxiliary winding by its backward revolvingfield;−−→

Es4 induced EMF in the backward branch of the auxil-iary winding by the backward revolving field of themain winding.

r1 , r2 resistances of the main and auxiliary windings, re-spectively;

X1 ,X2 leakage reactances of the main and auxiliary wind-ings, respectively;

rr rotor resistance (referred to the main winding);Xr rotor leakage reactance (referred to the main

winding);Xm magnetizing reactance;s rotor slip.

Referring to Fig. 4 and applying KVL for both the main andauxiliary windings gives

V1 = I1 (Z1 + Zf 1 + Zb1) + −−→Es1 + −−→

Es2 (8)

V2 = I2 (Z2 + Zf 2 + Zb2) + −−→Es3 + −−→

Es4 (9)

where Z1 is the series impedance of the main winding, which isgiven by

Z1 = r1 + jX1 . (10)

Z2 is the series impedance of the auxiliary winding, which isgiven by

Z2 = r2 + jX2 (11)

and

Zf 1 = 0.5 × jXm (rr/s + jXr )rr/s + j (Xr + Xm )

(12)

Zf 2 =Zf 1

a2 (13)

Zb1 = 0.5 × jXm (rr/(2 − s) + jXr )rr/(2 − s) + j (Xr + Xm )

(14)

Zb2 =Zb1

a2 . (15)

with Zb1 and Zb2 being the equivalent series impedances ofthe backward branches of the main and auxiliary windings,respectively.

Referring to Fig. 4, the following relationships can be writtenas

−−→Es1 = −ja

−−→Ef 2 = −jaI2Zf 2 = −jaI2

Zf 1

a2 (16)

−−→Es2 = ja

−→Eb2 = jaI2Zb2 = jaI2

Zb1

a2 . (17)


Fig. 5. Equivalent circuit of the unsymmetrical two-phase IM when operatedunder symmetrical operating conditions.

Hence, (8) can be rewritten as

V1 = I1 (Z1 + Zf 1 + Zb1) − jaI2Zf 1

a2 + jaI2Zb1

a2 . (18)

Similarly, (9) can be rewritten as

V2 = I2 (Z2 + Zf 2 + Zb2) + jI1Zf 1

a− j

I1Zb1

a. (19)

Substituting I2 = jaI1 [which are the symmetrical operatingconditions given by (4) and (5)] in (18) and (19) gives

V1 = I1 (Z1 + 2Zf 1) (20)

V2 = I2 (Z2 + 2Zf 2) = I2

(Z2 + 2

Zf 1

a2

). (21)

Equations (20) and (21) are used to derive a simplified equiva-lent of the two-phase unsymmetrical motor when operated undersymmetrical operating conditions. This simplified equivalent isshown in Fig. 5.

IV. CONSTANT AIR-GAP FLUX OPERATION

The essence of the SFC scheme is to maintain the motorair-gap flux at its rated value at various operating conditions.Referring to Fig. 5, the magnetizing current of the main windingcan be written in terms of ω as

Im1 = I1rr + jsXr

rr + js (Xm + Xr )(22)

s =ωs − ωm

ωs=

ωsL

ωs(23)

whereωsL is the rotor slip frequency in radians per secondωs is the synchronous frequency in radians per second, andIm1 is the magnetizing current of the main winding.The magnitude of the current in the main windings |I1 | is

obtained by substituting (23) into (22) and rearranging

|I1 | = |Im1 |

√r2r + ω2

sL (Lm + Llr )2

r2r + ω2

sLL2lr

. (24)

Equation (24) presents the relationship between the rotor slipfrequency ωsL and the magnitude of the magnetizing current ofthe main winding |I1 |. In order to keep the air-gap flux of themain winding fixed at its rated value for various rotor slips, |Im1 |in (24) is kept constant at its full-load magnitude (|Im1 |f .l.). Asimilar expression can be obtained for the magnetizing currentof the auxiliary winding.

Fig. 6. Relationship between |I1 |, |I2 |, and ωsL for constant value of themotor magnetizing current.

Fig. 7. Unsymmetrical two-phase motor. (a) Auxiliary and main windings ofthe single-phase motor. (b) D–Q transformation of the motor.

With the magnitudes of the motor magnetizing currents keptfixed at their rated values for various operating conditions, themotor never enters saturation, and better utilization of the mo-tor is ensured. Thus, this control scheme resolves IM into anequivalent separately excited dc motor in terms of its speed ofresponse, and not in terms of decoupling of the flux and torquechannels [12].

The rated value of the motor magnetizing current of the one-phase operation (|Im1 |f .l.) is calculated using the motor param-eters given in Appendix A to give

|Im1 |f .l. = 1.71A. (25)

Substituting (25) into (24) results in

|I1 | = 1.71

√r2r + ω2

sL (Lm + Llr )2

r2r + ω2

sLL2lr

. (26)

Once again, the current in the auxiliary winding is given interms of the current in the main winding by (4). The relationshipbetween the magnitude of the current in the main winding andthe rotor slip is shown in Fig. 6. The current in the auxiliarywinding is obtained by multiplying (26) by the ratio “a.” Fig. 6is used in the look-up table shown in Fig. 8.

Fig. 6 shows that a limit can be set on the maximumvalue of the main and auxiliary winding currents by limitingthe maximum excursion of the rotor slip frequency ωsL to a


Fig. 8. Schematic diagram of the proposed control scheme (P is the number of motor poles).

predetermined value. Limiting the maximum value of both themain and auxiliary winding currents has the advantage of reduc-ing the current ratings of the power inverter switching devices,thus reducing the overall system cost.

In this paper, the maximum value of the currents of the mainand auxiliary windings is set to twice their full-load values.

V. PROPOSED DRIVE SYSTEM OF THE UNSYMMETRICAL

TWO-PHASE IM

Fig. 8 shows the proposed control scheme. The scheme em-ploys the SFC strategy in order to realize single-phase IM drivesystem with high dynamic performance. It consists of an innercurrent loop and an outer speed-feedback loop.

The principle of operation of the control scheme is as follows.First, the actual motor speed ωm is compared with its referencesignal ωref to produce the slip frequency ωerr . The slip fre-quency ωerr is conditioned by the proportional–integral (PI)regulator and then passed through a limiter, which limits the ex-cursion of the slip frequency to a maximum predetermined value(35 rad/s, as indicated in Fig. 6) to produce ωsL . The value ofωsL is used along with the look-up table shown in Fig. 6 toobtain the magnitude of the commanding signal of the currentin the main winding (|I1 |). The magnitude of the commandingsignal of the current in the auxiliary winding (|I2 |) is obtainedby multiplying |I1 | by the ratio “a.”

Second, ωsL is added to the motor actual speed ωm to pro-duce ω. The variable ω is then multiplied by the number ofmotor pole pairs to produce the inverter output frequency ωs .The values of ωs along with |I1 | and |I2 |are used to determine

the reference waveforms of the current in the main (i1,ref ) andauxiliary windings (i2,ref ), respectively.

Finally, the actual stator currents are compared with their re-spective reference waveforms, i1,ref and i2,ref , in a hysteresiscurrent comparator. The output of the hysteresis comparator ofthe auxiliary winding is used to control the inverter switchingdevices S1 and S2 . Likewise, the output of the hysteresis cur-rent controller of the main winding is used to control the inverterswitches S3 and S4 . Hence, the duration of the turn-ON/OFF inter-val of each of the four devices is modulated such that the error be-tween the actual motor speed and the reference speed is reduced.

VI. COMPUTER SIMULATION AND PERFORMANCE EVALUATION

The stator windings (main and auxiliary) of the unsymmetri-cal two-phase motor are orthogonal with a 90-electrical-degreephase-shift. Hence, they readily lend themselves to the D–Qaxis representation in the stationary frame of reference.

The rotor is a squirrel cage and represented by equivalenttwo coils transformed to the stationary D–Q axis, as shown inFig. 7(b). Since the two stator windings have different numbersof turns, they will yield different mutual reactances. Therefore,a transformation is made to refer the auxiliary winding to anequivalent winding with the same number of turns as that of themain coil. The voltage equations (in per unit) in the stationaryD–Q axis are given by (27)–(30) [13]:

v′2 = r′2i

′2 +

1ωs

•ψ′

2 (27)

v1 = r1i1 +1ωs

•ψ1 (28)


0 = rr ird +1ωs

•ψrd +

ωm

ωsψrq (29)

0 = rr irq +1ωs

•ψrq −

ωm

ωsψrd . (30)

The equations of motion are given by

Te =P

2ωs(ψ′

2i1 − ψ1i′2) (31)

•ωm =

P

2J(Te − TL − Tdamp). (32)

The stator and rotor currents are given by

ψ′2

ψ1

ψrd

ψrq

=

X ′2 + Xm 0 Xm 0

0 X1 + Xm 0 Xm

Xm 0 Xr + Xm 0

0 Xm 0 Xm + Xr

×

i′2

i1

ird

irq

(33)

where prime superscript denotes quantities referred to the main(Q-winding) winding, ψ′

2 and ψ1 are the flux linkages of theauxiliary (D) and main (Q) windings, respectively, ψrd and ψrq

are the D and Q rotor flux linkages, respectively, i′2 and i1 arethe instantaneous currents in the main and auxiliary windings,respectively, ird and irq are the D and Q rotor currents, respec-tively, Te, TL , and Tdamp are the developed, load, and dampingtorques, respectively, P is the number of poles, and

r′2 = a2 × r2 X ′2 = a2 × X2

v′2 = a × v2 i′2 = i2 ×

1a. (34)

Equations (30)–(34) are used to study the performance of theproposed drive system and obtain the simulation results usingMATLAB/Simulink [14].

The steady-state torques and the respective frequency spec-trum of both modes of operation, namely, the one-phase andtwo-phase symmetrical operation modes, are calculated using(27)–(33), and depicted in Fig. 9. It is evident from the figurethat the proposed two-phase symmetrical operation scheme sup-presses the double-frequency oscillations in the torque, which isa large component in the one-phase motor. The high-frequencycomponents of the torque (evident from the time response)are due to the power inverter switching action. These high-frequency components have little impact on the motor behaviorsince they will be filtered out by the combined inertia of the loadand the motor. It is also evident from the figure that the two-phase mode of operation has higher average developed torque.

The electric torque, output and input powers, and efficiencyare calculated as reported in Appendix B and depicted in Fig. 10.It is evident from this figure that the proposed scheme providesconsiderable improvement in the motor performance.

Fig. 11 shows the simulation results of the speed responseof the two-phase unsymmetrical motor. The parameters of the

Fig. 9. (a) and (b) Time and frequency response of the electric torque of theone-phase IM. (c) and (d) Time and frequency response of the electric torque ofthe same IM when operated as a symmetrical two-phase IM. (a) One-phase IMtime response. (b) One-phase IM frequency response. (c) Two-phase IM timeresponse. (d) Two-phase IM frequency response.

Fig. 10. (a) Torque–slip characteristics. (b) Output power versus slip.(c) Efficiency versus power.

PI regulator of the outer feedback-loop speed are as follows:Kp = 1 and Ki = 0.250 (Appendix C). The hysteresis windowsof the hysteresis current controllers of the currents in the mainand auxiliary windings are chosen to be 10% of the maximumvalue of its respective full-load value. Simulation results showthat the scheme has been successful in driving the motor fromstandstill to full load in 0.4 s and its steady-state speed error is1.7%. Fig. 12 shows that the scheme is successful in limitingthe starting current of the motor windings to approximately amaximum of 2 per unit (p.u.) of the full-load value of eachwinding.

VII. EXPERIMENTAL RESULTS

Experimental verification of the proposed scheme was carriedout by constructing an experimental setup in the laboratory. Theexperimental setup was built using a rapid control prototypingdevelopment system, which employs DS1104 by dSPACE [15].

Figs. 13 and 14 show photographs of the experimental setupthat comprise the following.


Fig. 11. Simulation results of the motor speed.

Fig. 12. Simulation results of the current in the main and auxiliary windings.

Fig. 13. Photograph of the experimental setup (front view). (1): computermonitor with Simulink program; (2): connector panel by dSPACE; (3): gatedrive circuit by SEMIKON; (4): power inverters; and (5): host computer withDS 1104 board inside.

1) A dSPACE DSP board DS 1104 plugged into the moth-erboard of a host computer: The DS 1104 performs allreal-time control functions, while the processor of the hostcomputer performs downloading of the control program,data logging, and data communications. It is worth men-tioning that all experimental results recorded in this paper

Fig. 14. Photograph of the experimental setup (backside view). (6): DC capac-itors; (7): single-phase IM operated as a two-phase motor; (8): DC dynamometer;(9): torque meter; and (10): load.

Fig. 15. Experimental results of the motor transient response at full load.

have been collected using the built-in data-acquisition ca-pabilities of the DS 1104. These data were later plottedusing MATLAB package [14].

2) Two single-phase half-bridge inverters: The bridge invert-ers are realized using SEMIKRON giant transistor mod-ules [16]. The power transistor module employed containsfour insulated gate bipolar transistors (IGBTs), each ofwhich can handle 40 A (rms) at 1200 V.

3) Two current sensors employing printed circuit board(PCB) mount Liaisons Electroniques-Mecaniques(LEM’s) Hall effect transducers: These current trans-ducers were used to sense the currents in the main andauxiliary windings.

4) A single-phase motor with quadrature encoder (with 1024pulses per revolution) fitted on the motor shaft to sense themotor speed for feedback: The motor parameters are givenin Appendix A.

5) The load realized by mechanically coupling a dynamome-ter to the shaft of the two-phase IM: The output of the dcgenerator was connected to a resistive load.

Fig. 15 shows the transient speed response with the motorfully loaded. The figure also shows that the control scheme is


Fig. 16. Motor steady-state speed and slip.

Fig. 17. Motor response to 100% step in the speed command.

Fig. 18. Motor response to triangular speed command.

capable of bringing the motor from standstill to full load inapproximately 0.4 s with steady-state speed error of 1.3% (seeFig. 16). It can be seen that both the simulation results (shownin Fig. 11) and the experimental results are in good agreement.The motor efficiency was measured to be 84%. This amountsto an efficiency gain of 8% compared to that when the motor isoperated as a single-phase motor with the same full-load value.

Fig. 17 shows the motor response to ±100% step in the speedcommand. The figure shows that the proposed scheme is capableof driving the motor in both clockwise and counterclockwisedirections with fast response. The figure also shows that thescheme is successful in limiting the starting current of the motorwindings to approximately a maximum of 2 p.u. of the full-loadvalue of each winding.

Fig. 18 shows the motor response to triangular speed com-mand. The figure shows that the motor speed can track its refer-ence signal with very fast response and small steady-state error.Figs. 15–18 show that the proposed control scheme results in ahigh-dynamic-performance drive.

VIII. CONCLUSION

A high-dynamic-performance symmetrical two-phase IMdrive system has been realized using an off-the-shelf single-phase IM. The proposed drive system employs the slip-frequency scheme to independently control the currents in thetwo stator windings of the single-phase IM. In addition to con-tinuous speed control and an efficiency gain of 8%, the proposedscheme provides the following cost-effective advantages.

1) The rating and, consequently, the cost of the inverter are re-duced. The control scheme independently limits the start-ing currents in both windings to predetermined values.

2) Notably, the proposed scheme achieves reduction of in-herent torque pulsations in the unsymmetrical two-phaseIM, and hence, endows the unsymmetrical motor with adesirable natural feature of its symmetrical counterpart.

3) The scheme does not require a powerful microcon-troller/microprocessor for its implementation.

The proposed control strategy can be potentially extended tothree-phase motor drives when the three-phase IM loses one ofits windings.

APPENDIX A

Single-Phase IM Parameters

Output power = 1/3 hp, four poles, N1/N2 = a = 1.10, andη = 76%.

Rated voltage = 115.00 V (rms) and supply frequency =50.00 Hz.

X1 = 2.22 Ω, r1 = 1.80 Ω, X2 = 2.80 Ω, and r2 = 4.10 Ω.Xr = 2.22 Ω, rr = 2.01 Ω, and Xm = 40.22 Ω.Rotational losses = 56 W and full-load slip = 5.0%.Vdc = 170 V (115

√2).

APPENDIX B

Calculation of Steady-State Quantities of the Two-Phase Modeof Operation

For a certain slip “s,” the main and auxiliary currents areobtained according to (20) and (21). The rotor copper lossesare calculated using the equivalent circuit shown in Fig. 5 asfollows:

Pcur = I2r1Rr + I2

r2Rr . (35)


The motor air-gap power is given by

Pg = I2r1

Rr

s+ I2

r2Rr

s. (36)

The electric torque is

Te =Pg

ωs. (37)

The motor output power is given by

Pout = Pg − Pcur − Prot . (38)

The input power is given by

Pin = Pg + I21 R1 + I2

2 R2 . (39)

APPENDIX C

Effect of Controller Parameters on System Performance

TABLE IEFFECT OF Kp WITH ki = 0

TABLE IIEFFECT OF Ki WITH kp = 1

REFERENCES

[1] L. Mhango and G. Creighton, “Novel two-phase inverter-fed inductionmotor drive,” Proc. Inst. Electr. Eng., vol. 131, no. 3, pp. 99–104, May1984.

[2] E. R. Collins, Jr., H. B. Puttgen, and W. E. Sayle, II, “Single-phase in-duction motor adjustable speed drive: Direct phase angle control of theauxiliary windings supply,” in Proc. IEEE Ind. Appl. Soc. Annu. Meeting,1988, pp. 246–252.

[3] D. Jang, G. Cha, D. Kim, and J. Won, “Phase-difference control of 2-phaseinverter-fed induction motor,” in Proc. 20th IEEE Power Electron. Spec.Conf., Jun. 26–29, 1989, vol. 2, pp. 571–578.

[4] D.-H. Jang and J.-S. Won, “Voltage, frequency, and phase-difference anglecontrol of PWM inverter-fed two-phase induction motor,” IEEE Trans.Power Electron., vol. 9, no. 4, pp. 377–383, Jul. 1994.

[5] M. Correa, C. Jacobina, A. Lima, and E. Silva, “Induction motor drivesystem for low-power application,” IEEE Trans. Ind. Appl., vol. 35, no. 1,pp. 52–60, Jan./Feb. 1999.

[6] M. Correa, C. Jacobina, A. Lima, and E. Silva, “Rotor-flux-oriented con-trol of a single-phase induction motor drive,” IEEE Trans. Ind. Electron.,vol. 47, no. 4, pp. 832–841, Aug. 2000.

[7] M. Correa, C. Jacobina, A. Lima, and E. Silva, “Vector control strate-gies for single-phase induction motor drive systems,” IEEE Trans. Ind.Electron., vol. 51, no. 5, pp. 1073–1080, Oct. 2004.

[8] N. Abdel-Rahim and A. Shaltout, “Operation of single-phase inductionmotor as two-phase motor,” in Proc. 28th Annu. Conf. IEEE Ind. Electron.Soc. (IECON), Sevilla, Spain, Nov. 5–8, 2002, pp. 967–972.

[9] D. G. Holmes and A. Kotsopoulos, “Variable speed control of single-phase and two-phase induction motors using a three-phase voltage sourceinverter,” in Conf. Rec. 1993 IEEE Ind. Appl. Soc. Annu. Meeting, vol. 1,pp. 613–620.

[10] W. H. Yeadon and A. W. Yeadon, Handbook of Small Electric Motors.New York: McGraw-Hill, 2001 (ISBN 007072332X).

[11] B. S. Guru and H. R. Hiziroglu, Electric Machinery and Transformers.New York: Oxford Univ. Press, 1995.

[12] R. Krishnan, Electric Motor Drives: Modeling, Analysis and Control.Upper Saddle River, NJ: Prentice-Hall, 2001, p. 350 (ISBN 0-13-0910147).

[13] C.-M. Ong, Dynamic Simulation of Electric Machinery Using MAT-LAB/SIMULINK. Englewood Cliffs, NJ: Prentice-Hall, 1998.

[14] MATLAB. (2004). [Online]. The Mathworks, Inc., Natick, MA. Available:http://www.mathworks.com/

[15] (2004). [Online]. Available: www.dspace.de[16] (2004). [Online]. Available: www.semikorn.com

Naser M. B. Abdel-Rahim received the M.Eng. andPh.D. degrees from Memorial University of New-foundland, St. John’s, NF, Canada, in 1989 and 1995,respectively.

From 2000 to 2005, he was an Assistant Profes-sor in the United Arab Emirates University (UAEU),where he received several research projects fundingfrom the UAEU as well as from ABB. He is currentlyan Associate Professor in the Department of Elec-trical Engineering, Benha University, Benha, Egypt.He is the author or coauthor of numerous publica-

tions in international journals and refereed conferences, where he also served asreviewer.

Adel Shaltout received the B.Sc. and M.Sc. de-grees from Cairo University, Giza, Egypt, the M.Sc.degree from McMaster University, Hamilton, ON,Canada, and the Ph.D. degree from the University ofSaskatchewan, Saskatoon, SK, Canada.

He was a Visiting Professor at several universities.He is currently a Professor of electrical machines atCairo University. His current research interests in-clude electric machines, power systems, and renew-able power energy.

608 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. … Shoubra/Electrical Engineering/916...Open-loop speed control of the two-phase IM using two single-phase half-bridge inverters operated

Documents