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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 9, NO. 4, JULY 1994 311

Voltage, Frequency, and Phase-DifferenceAngle Control of PWM Inverters-FedTwo-Phase Induction MotorsDo-Hyun Jang, Member, IEEE, and Jong-Soo Won

Abstracf- A phase-difference angle (PDA) controlledpulsewidth-modulated (PW M) inverter is proposed foradjustable speed drive for the two-phase induction motors.Output waveforms are fixed over the whole operating rangeof the motor. The motor torque is controlled not by themod ulation of the phase voltage, but by the PD A. Based onthe selected harmonic elimination (SHE) PWM technique,the commutation angles of the output voltage are calculated.Several characteristics of the two-phase induction motor drivenby the PDA inverter are analyzed. Second, a hybrid PWMinverter is proposed also to compensate for the degradationof the efficiency at small PDA. Not only the PDA but also thevoltage amplitude and frequency are used as the p arameters forcontrolling the torque of the motor in the hybrid inverter. Thespeed chara cteristics of the two-phase ind uction motor driven bythe hybr id PWM inverter are more flexible than when the m otorin driven by the conven tional PWM in verter, which requ iresadjustable comm unication angles.

I. INTRODUCTIONMALL two-phase induction motors had been popular inS ervomechanisms since the 1940's. In general, the speedcontrol for two-phase induction motors is carried out byadjusting the control winding voltage by the quadrature controlsignal of an AC amplifier while exciting the reference windingvoltage with constant frequency and amplitude. This typicalcontrol method had been widely used because an AC sourceis usually available and an AC amplifier s drift-free. However,the drift ability of an AC amplifier is not satisfactory.The advent of low-cost power semiconductor devices inthe early 1960's solved the drift problem of DC source, soin servomechanisms, the two-phase induction motors werereplaced with DC motors, which are easy to control and havea high efficiency. But DC motors have some disadvantages,such as the friction between the commutator and brush,which causes high maintenance cost and shortens work life.Therefore, AC motors are being studied again to reducemaintenance cost and to increase work life.The ratings of two-phase induction motors are usually low,so simple inverters are needed. Although the two-step invertersreported earlier we simple and cheap 111, [2], the harmoniccontents of output voltage are relatively high, and, with a fixed

Manuscript received March 1, 1992; revised March 22, 1994.D.-H. Jang is with the Department of Electrical Engineering, HoseoJ.-S. Won is with the Department of Electrical Engineering, Seoul NationalIEEE Log Number 9404409.

University, Asan-Kun, Chungnam, Republic of Korea.University, Seoul, Republic of Korea.

DC voltage source, the amplitude of output voltage cannotbe controlled. Hence, the motor speed control capability isweak, and the motor efficiency at slow speed is low becauseof magnetic saturation.The conventional pulsewidth-modulated (PWM) invertersrepresent the best solution for the reduction of the harmoniccontent of the output voltage. However, the application tothe two-phase induction motors has been restricted, becausethe drives are too complicated for small two-phase inductionmotors [l].In this paper, a new type of inverter, the phase-differenceangle (PDA) controlled PWM inverter, is proposed for ad-justable speed drive for small two-phase induction motors.Instead of the voltage amplitude and frequency, the PDAbetween two winding voltages is used as the parameter forcontrolling the motor torque. Based on the selected harmonicelimination (SHE) PWM technique, the commutation anglesof the output voltage are calculated. Several characteristics ofthe two-phase induction motor driven by the PDA-controlledinverter are analyzed.Second, a hybrid PWM inverter, which combines the PDAcontrol and the voltage control, is proposed to compensatethe degradation of the efficiency at small PDA. Not only thePDA but also the voltage amplitude and frequency are usedas parameters for controlling the motor torque in the hybridPWM inverter. Several characteristicsof two-phase inductionmotor driven by the hybrid PWM inverter are analyzed.

11. TORQUEONTROL OF TWO-PHASENDUCTION MOTORSThe theoretical methods to control the torque of two-phase1) Voltage control: This method can be divided into twotypes. First, the voltage vb supplied to the winding ofphase B may vary from 100 to 0 [%I, while the voltage

V, supplied to the winding of phase A remains constant,and the PDA is fixed at 90". Second, the voltagessupplied to the two windings may vary together, whereasthe amplitudes of the two winding voltages are equalto each other, and the PDA is fixed at 90". Frequencycontrol should be taken into account as the voltages varyin the second method.2) PD A control: The PDA between two winding voltagesmay vary from +90 to -goo, whereas the amplitudesof two voltages are fixed and identical to each other.

induction motors are as follows [3].

0885-8993/94$04.00 0 994 IEEE

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378 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 9, NO. 4, JULY 1994

Fig. 1. Schematic representation of two-phase induction motor.

Fig. 2.anced condition (in terms of phase A ) .Equivalent circuit for two-phase induction motor under the unbal-

The first method of (1) has been used in conventional torquecontrol of two-phase induction motors, and the second methodof ( 1 ) has rarely been applied to two-phase induction motorsby means of recently developed inverters [4] . However, thePD A control method of (2) has not been applied until now.A. Torque Formula

Fig. 1shows the schematic representation of two-phase in-duction motors, where the windings of phase A are located 90'from the windings of phase B in space, and the impedancesof the two windings are identical. It is assumed that thefundamental voltages U , , U b are supplied to the two phasewindings, as follows:

where K(=Vb/V,) is voltage ratio, and w i is synchronousangular velocity, and 4 is the PDA between two windingvoltages. If V, and Vb are identical and 4 is fixed at go", theseexpressions represent the two balanced winding voltages in thevoltage control wherein the motor torque is controlled by thevoltage amplitude and frequency. This method is regarded asa balanced operation.If V is constant and r#~ is fixed at go", these expressionsrepresent the two winding voltages in a typical servo systemof two-phase induction motors wherein the motor torque iscontrolled by the voltage ratio K . On the other hand, if V, andv b are constant and identical, these expressions represent thetwo winding voltages in the PDA control wherein the motortorque is controlled by the PDA between two winding voltages.These two methods are regarded as an unbalanced operation.The torque of the two-phase induction motors under unbal-anced operation can be analyzed by a symmetrical coordinatemethod invented by Forescue [ 5 ] . Fig. 2shows the equivalent

Fig. 3. Vector diagram of two unbalanced winding voltages.

circuit of phase A winding of the two-phase induction motorunder unbalanced conditions. The phase voltage is composedof the forward voltage V& and backward voltage V a b , whichcorrespond to the forward and backward revolving field,respectively. Similarly, the equivalent circuit of phase B hasthe forward voltage Vbf and backward voltage v b b . As shownin Fig. 3, the relationship between the forward voltages V,fand Q,ff two phases is balanced as V b f = jV , f . The sameis true for the backward voltages, that is, v b b = - . jvab. Thetwo revolving fields rotate in opposite directions with respectto each other. Thus, the two winding voltages V, and Vb areexpressed in terms of V,f and V&,as follows:

Therefore, if V, and V b are known, Vaf and V a b can beexpressed as follows:

The torque generated by each component can be easilyevaluated. Their algebraic sum becomes the net intemal torque,and is expressed as follows:

where V,fl and Vabl represent the amplitude of the voltageinduced by the forward revolving field and by the backwardrevolving field, respectively, and their squares are expressedas follows:

where

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JANG AND WON: PW M INVERTERS-FED TWO-PHASEINDUCTION MOTORS

PHASE A

PHASE B

379-- w-. wt

I I I I I I I I

-.j6

SPEEDp.u.

SPEEDp.u.

SPEEDp.u.

(c)Fig. 4. Speed-torque characteristics of two-phase induction motor underseveral conditions. (a) Balanced voltage control (l'/f = 5/3). (b) Unbalancedvoltage control. (c) PDA control.

A f ( K , ) = (1 + K 2+ 2Ksin$)/4A b ( K , 4 >= (1 + K 2 - 2Ksin4)/4. (12)(13)

Substitution of (8) and (9) into (7) yields the followingequation:

The first term in (14) shows the positive component of thetorque generated in two-phase induction motors, and thesecond term shows the negative component.

I I I I I I I I0 90 180 27 0 360 450 540 630 72 0(b)

Fig.5. Waveforms of the output voltages of two-phase inverters. (a)Two-step inverter. (b) PDA-controlled inverter.

B . Torque Characteristics of Two-Phase InductionMotors under Unbalanced ConditionsUsing (14), the torque generated in two-phase inductionmotors under several control methods is analyzed on a testmotor 03 = 2, T I = 11 7 R , z1 = 125 R , x, = 828 R,

TZ = 51 7 R, x2 = 15.6 0, f = 60 Hz , V, = Vb = 10 0V) . Fig. 4 shows the speed-torque curves of the test motorunder several control methods. Torque characteristic under thebalanced voltage control is shown in Fig. $(a), in which thetwo winding voltages and frequency vary under the constantvolthertz ratio (V/f 5/3) , and $ is fixed at 90'. Torquecharacteristic under the unbalanced voltage control is shownin Fig. 4(b), in which the voltage of phase B is vanedand the voltage and frequency of phase A is fixed. Torquecharacteristic under the PDA control is shown in Fig. 4(c), inwhich 4 is varied and the two winding voltages are fixed.The PDA control of Fig. 4(c) shows several features in itstorque characteristic. First, for a negative phase angle, themotor can be made to operate in quadrant I11 to where theother methods cannot reach. This feature of the PDA controlenables a two-phase induction motor to easily reverse therotating direction by changing the PDA. An excellent brakingcapability is another feature of the PDA control. The inherentcapability of the plugging operation can be exploited to shortenthe braking time to a standstill.

111. DRIVEDESCRIPTIONFig. 5 shows the main structure of the PWM inverter-fedtwo-phase induction motor. It consists of a two-phaseinduction motor, a half-bridge type inverter, a driving circuit,

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2Ed ET

I 7 1mHASE 0OTORFig. 6. PDA-controlled inverter-fed two-phase induction motor.

a fixed DC voltage source, and a microprocessor. In thetwo-step inverter-fed two-phase induction motor reportedearlier [l], [2], the square voltage waveforms with C$= 90,are supplied to the two windings, as shown in Fig. 6(a). Thisinverter can operate two-phase induction motors without amicroprocessor, because its topology is simple. The speed ofthe two-phase induction motor driven by two-step inverter iscontrolled by the frequency only. Therefore, as the frequencyis lowered, the magnetic flux in the core becomes saturated,which results in poor efficiency and weak speed controlcapability. In addition, the square voltage waveform hassignificant harmonics of low order.An excellent torque characteristic is expected if the PWMtechnique such as that used in three-phase VVVF inductionmotor drives is applied to two-phase induction motors. How-ever, it is undesirable, because the complex and expensivedrives are required for small two-phase induction motors.Therefore, this paper proposes a simple and inexpensive ad-justable speed drive of two-phase small induction motor.The PDA control method for controlling the torque of two-phase induction motor is applied to the proposed inverter,named the PDA-controlled inverter. Fig. 5(b) shows an outputvoltage waveform under PDA control that exhibits an arbitraryphase difference between the winding voltages. The windingvoltages are modulated based on the selected harmonic elim-ination (SHE) PWM technique [6], and their waveforms arenot subject to change over the whole operating range of themotor. The motor torque is controlled not by the modulationof the phase voltage, but by the PDA. The proposed inverteroffers the following advantages.

Since the commutation angles need not be changed,the hardware and/or software are relatively simple toimplement.In conventional adjustable-speed PWM drives, both theamplitude and frequency of the voltage should be ad-justed in order to meet the constant volthertz (V/f)requirement. In the PDA-controlled inverter, however,they are fixed throughout the operation, which furthersimplifies the implementation.The speed reversal can easily be carried out by simplychanging the PDA without any auxiliary device.

t

I

Harmonic numberFig. 7.analysis ( M = 8) .PWM waveform by the SHE PWM technique and its harmonic

In the PDA-controlled inverter, however, the motor effi-ciency decreases as the PDA approaches 0. This shortcomingis analyzed in detail in Section V, and its compensation isdiscussed in Section VI.IV. IMPLEMENTATIONF SH E PWM

The SH E PWM technique [6] for calculating the commu-tation angles of the output voltage is mathematically the bestin eliminating harmonics. For application of the SHE PWMtechnique, M commutation angles in 1/4 cycle should bedetermined, where M is the number of specified harmonicsto be eliminated. It yields M nonlinear equations, which aresolved by the Newton-Raphson iteration method. The PWMwaveform for M = 8 is selected in this paper, as shown inFig. 7. The harmonics up to the seventeenth (2M + 1) orderare eliminated, and the fundamental component remains nearly1. The dominant harmonics are nineteenth, twenty-first, andtwenty -third.The switching angles are stored in a ROM. It requiressuch a small capacity of ROM that can be stored, withPDA control software, in a built-in memory of a one-chipmicroprocessor. Fig. 8(a) and 8(b) show the experimentaloutput voltage waveforms of the PDA-controlled inverter. Theharmonic spectrum of the output voltage is shown in Fig. 8(c).The shape of PWM waves remains unchanged, even thoughthe PDA varies and harmonic spectrum nearly agrees withthat of Fig. 7.

v. ANALYSISF MOTORCHARACTERISTICSSeveral characteristics of a test motor driven by the PDA-controlled inverter are shown in Fig. 9 and Fig. 10. In eval-uating analytical characteristics, it is assumed that the effectsof the magnetic saturation of the core and the harmonics ofthe output voltages are negligible. Fig. 9(a) and 9(b) show

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JANG A N D WON: PW M INVERTERS-FED TWO-PHASE INDUCTION MOTORS 38

LOAD TORQUE N.cm(a)

0 0.8 1.6 2.4

Fig. 8. Experimental PW M waveforms and harmonic analysis. (a ) Ato = 90'. (b ) At o = -15'. (c) Harmonic analysis.

the variation of several characteristics as a function of loadunder the balanced (4 = 90") and unbalanced operations(4 = 45"), respectively. In both cases, the measured char-acteristics closely coincide with the calculated ones. It shouldbe noted that in the unbalanced operation, the motor speed andefficiency are lower and the power input is higher than thosein the balanced operation, as expected.Fig. 10 shows a comparison of the characteristics of thetest motor driven by the PDA-controlled inverter and by thetwo-step inverter under a fixed load (T' = 0.9 N.cm). Themotor efficiency and speed increases and the power inputdecreases as the control parameter increases. Note that therightmost operating conditions in each figure are the same.When the motor is driven by the PDA-controlled inverter,the power input is lower, and the efficiency is higher thanwhen it is driven by the two-step inverter. While the PDA

While the PDA-controlled inverter has several advantages,the motor efficiency becomes poor, although it is still betterthan two-step inverter drive for a small PDA. In this sec-tion, the hybrid PWM inverter, which combines the PDAcontrol method and the voltage control method, is proposedto compensate the degradation of the efficiency. Not only thePDA but also the voltage amplitude and frequency are used asthe parameters for controlling the torque in the hybrid PWMinverter drive.In order to decrease the motor speed, the PDA has to bemade small, as shown in Fig. 1I(a), which causes a poorefficiency. Instead, if the voltage amplitude and frequency arereduced with a larger PDA, as shown in Fig. 1I(b), the motorefficiency becomes high at low speed.In implementing the hybrid PWM inverter operation, thetorque-speed plane is divided into several sections. In eachsection, the voltage amplitude and frequency are set to beconstants with which the commutation angles are calculatedby the SH E PWM technique. During th e operation, the PWMwaveform is fixed as long as the operating point does not

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6 1

E --20

0

EFRCIENCY

50 60 go

E 2400

W 120020

YPUT SPEED /"

o i o io io 40 so 60FREQUENCY Hz

Ib)

SPEED p.u.(a)

SPEED p.u.(b)

,-> Fig. 12.sections. (a) constant/f .Speed-torq ue Characteristicsof two-phase induction motor with fiveFig. 10. Several characteristicsof two-phase induction motor under the f ixedload (TL 0.9N.cm). (a) The PDA-controlled inverter. (b) WO-step inverter. Variable v/f.

PHASE A t

PHASE B tr IU U I I I UUI L I . . ~ I U U I I I u u u

PHASE A t

PHASE B tU uu U U U u u u u u u u u U U U I

(b)Fig. 1 1 . PWM waveforms in hybrid PWM inverter. (a) The rated voltageand the narrower PDA. (b) The lower the voltage and the wider the PDA.move into another section, and the torque of the motor iscontrolled by the PDA. With given voltage and frequency. Theoperating range of the PDA in each section can be determinedby considering the maximum torque of each section.

Fig. 12 shows the speed-torque curves of the test motordriven by the hybrid inverter. In Fig. 12(a), the solid lines aretorque curves at 4 = 90"for five different voltage levels wherethe frequency for each curve is set to have constant voltsbertzratio (V/f 5/3 V/Hz). The number of sections is arbitrarilychosen. These curves devide the torque-speed plane into fivesections, from section [I] to section [VI. The section [VI, forexample, is between the uppermost curve and the next curve.Over this section, the voltage and frequency are maintainedas they are in the uppermost curve. The operating conditionsfor each section are given as follows: [I] voltage V = 20V, frequency f = 12 Hz, he operating range of the PD A0 5 4 5 90' [11] 40 V , 24 Hz, 22.5' 5 4 5 90' [111] 60 V ,36 Hz, 32' 5 4 5 90' [IV] 80 V , 48 Hz, 37.5' 5 4 5 90'[VI 100 V, 60 Hz, 41.5' 5 4 5 90'.The operating range of the PD A for each section listedabove are evaluated so that the variation of the torque curvecan cover all of the operating range. But it poses someproblems. For example, suppose the inverter operates on 100V , 60 Hz, and 4 = 90". Then the torque curve becomesthe uppermost one in the figure. Under this condition, as 4is decreased from 90' to a smaller value, the torque curvemoves from the uppermost curve toward the next solid line.When the PDA is decreased to the value that yields no loadspeed, the same as that of the second solid line, the torquecurve falls on the broken line near the second solid line.Obviously, there is some overlap between the lower portion

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38 3ANG AND WON: PWM INVERTERS-FED TWO-PHASE INDUCTION MOTORS

.-r20

INPUTFig. 13. Several characteristic s of two-phase induction motor driven by thehybrid PWM inverter.

of the section [VI and the upper portion of the section [IV].Generally, the solid line that indicates the torque curve of asection at (b = 90 does not coincide with the broken line thatindicates the torque curve at the smallest PDA of the adjacentsection. This may cause a problem in speed control when thesection is switched from one to another, because of an abruptchange of the torque-speed characteristic of the motor.In order to avoid the abrupt change of the torque at theboundary, the voltage, frequency, and operating range of thePDA in each section are adjusted appropriately. The result isshown in Fig. 12(b), where two limit curves at the boundaryof adjacent sections approach each other very closely. Themodified values for Fig. 12(b) are as follows: [I] V = 35 V,f = 46 Hz, 0 5 4 5 90 II] 53 V, 51 .5 Hz, 28 5 4 5 90,[111] 70 V, 55.5 Hz, 31 5 4 5 90, [IV] 85 V, 58 Hz,42 5 4 5 go, [VI 100 V, 60 Hz, 5 5 (b 5 90. Notethat the V/f atios in each section are kept to be less thanthe rated ratio.The motor characteristics-efficiency, power input, poweroutput, and speed-under a constant load are shown inFig. 13, which is based on Fig. 12(b). From the figure, it isobserved that the motor efficiency becomes higher, and thespeed characteristic is nearly linearized and continuous at theboundary. It is expected that the motor performance will beimproved if the number of sections are increased.

VII. CONCLUSIONThe PDA-controlled PWM inverter is proposed for theadjustable speed drives of two-phase induction motors. Outputwaveforms are not subject to change over the whole operatingrange of the motor. The motor torque is controlled not by themodulation of the phase voltage, but by the PDA. Based onthe SHE PWM technique, the commutation angles of outputvoltage are calculated for eliminating the harmonics.The proposed inverter has the following advantages. Thecommutation angles of the PWM output voltages are fixed, sothe hardware and software of the PDA-controlled inverters aresimple. Frequency control is not taken into account, becauseof the fixed output voltages. Speed reversal can easily be done

by changing the PDA. The efficiency of two-phase inductionmotor driven by the PDA-controlled inverter is superior to thatof two-step inverter, but it is low at small PDA.Second, the hybrid PWM inverter, which combines thePDA control method and voltage control method, is proposedto compensate the degradation of the efficiency at smallPDA. Not only the PDA but also the voltage amplitudeand frequency are used as the parameters for controllingthe motor torque in the hybrid PWM inverter. Hence, themotor efficiency is improved compared with that of the PDA-controlled inverter. Also, the speed characteristics are moreflexible than when the motor is driven by the conventionalPWM inverter, which requires adjustable commutation angles.It is expected that the motor performance will be improved ifthe number of sections are increased.

REFERENCESL. M. C. Mhango and G. K. Creighton, Novel two-phase inverter-fedinduction motor, Proc. IEE, vol. 131-B, no. 3, pp. 99-104, 1984.G. Rojat, J.L. Mertz, and A. Foggia, Theoretical and experimentalanalysis of a two-phase inverter-fed induction motor, IEEE Trans. Ind.App., vol. IA-15, no. 6, pp. 601-606, 1979.D. Connelly, The two-phase induction motor used as a servo motor,in proc IEE, pp. 366-374, Aug. 1960.D. Alexa, Static frequency converter for supplying an asynchronoustwo-phase motor, Pror. IEE, vol. 134-B, no. 1, 1987.C. L. Fortescue, Method of symmetrical co-ordinates applied to thesolution of polyphase networks, AIEEE Trans., pp. 1027-1032, 1918.H. S . Patel and R. G. Hoft, Generalized techniques harmonic elim-ination and voltage control in thyrister inverters: Part I-Harmonicelimination, IEEE Trans. Ind. Ap p., vol. IA-9, pp. 310-317, 1973.

D.-H. Jang (S87-M90) received the B.S. degreefrom Hanyang University, Seoul, Republic of Ko-rea, in 1980, and the M.S. and Ph.D. degrees fromSeoul National University, Seoul, Republic of Ko-rea, in 1982 and 1989, respectively.Since 1985, he has been a Lecturer and an As-sistant Professor with the Department of ElectricalEngineering, Hoseo University, Chungnam, Repub-lic of Korea, where he is currently an AssociateProfessor. His research interests are in the area ofcontrol of motor drives, PWM technique, and acchopper.

J.4. on received the B.S. degree from SeoulNational University. Seoul, Republic of Korea, in1952.From March 1964 to August 1993, he was withthe Department of Electrical Engineering, Seoul Na-tional University, Seoul, Republic of Korea, wherehe taught electrical machine control and design. Heis currently a Professor Emeritus at Se oul NationalUniversity.Mr. Won has published a number of papers onelectric machine control and two books on electricequipment. He worked on a technical co mmittee in the international vocationalcompetition from 1968 to 1970. Hi s main research field is electrical machines.In particular, he is interested in the design of fractional horsepower electricalmotors, permanent magnet ac motors, and microprocessor-based small motordrives.