Problems Incurred in a Vector-Controlled Single-Phase Induction ...

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 1, JANUARY 2013

Problems Incurred in a Vector-Controlled Single-Phase Induction Motor, and a Proposal for a Vector-Controlled Two-Phase Induction Motor as a Replacement Do-Hyun Jang, Member, IEEE

Abstract—This paper presents several of the problems encountered with vector-controlled single-phase induction motor (SPIM), and discusses about the complex implementation of a vector controlled SPIM drive. The vector-controlled symmetrical two-phase induction motor (TPIM) is presented as a viable replacement for the vector-controlled SPIM. The implementation of the proposed vector-controlled TPIM is simple compared to the vectorcontrolled SPIM. All the TPIM parameters can be calculated simply and precisely. The proposed strategy for TPIM is derived from the indirect vector control strategy used for three-phase ac machines. Several differences between the vector control strategies for the TPIM and for three-phase ac motor are discussed. The validity of the proposed vector-controlled TPIM was verified by simulations and experiments. Index Terms—Vector-controlled single-phase induction motor (SPIM), vector-controlled two-phase induction motor (TPIM), unsymmetrical motor.

I. INTRODUCTION INGLE-phase induction motors (SPIMs) are employed widely in the fractional power range, particularly in households where a three-phase ac electrical supply is not available. SPIMs are used typically to maintain a constant speed, for example, in fans and vacuum cleaners. SPIMs normally require auxiliary winding and main winding as well as a capacitor to produce the starting torque. SPIMs are typically classified according to their starting technique: split-phase motor, capacitor (capacitor-run or capacitor-start) motor, and shaded-pole motor [1]. Fig.1 shows typical speed control of SPIMs by voltage control using the power switches. Fig. 1(a) shows a triac-controlled strategy that adjusts the supplied voltage to the capacitor motor according to the load conditions. The capacitor in a capacitor motor causes an auxiliary phase current to lead the main phase voltage, creating a large angle of displacement between the currents in the two windings. This strategy can only control the speed of the capacitor motor over a narrow range owing to its fixed synchronous frequency. Since SPIMs are used in a range of

S

Manuscript received March 9, 2012; accepted April 24, 2012. Date of current version September 11, 2012. Recommended for publication by Associate Editor K. M. Ralph. The author is with Hoseo University, Chungnam, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2012.2199772

Fig. 1. Power converter for an SPIM drive. (a) AC voltage controller by triac. (b) Two-phase half-bridge inverter.

applications at low power levels, the necessity of higher power efficiency and a workable adjustable speed control are of great importance [2]. Fig. 1(b) shows a two-leg inverter connected to an SPIM. When this inverter is used, the speed of the SPIM can be controlled over a wide range by adjusting the frequency. Several studies have concentrated on vector control strategies for SPIMs (as “vector-controlled SPIM”) [4]–[10] because they are used widely. On the other hand, the SPIMs are classified as unsymmetrical two-phase induction motors (TPIMs) because the parameters of the main and auxiliary windings are not identical. Therefore, when the SPIM is operated by the vector control strategy, significant problems are generated due to unbalanced operation. This paper presents several problems encountered in conventional vector-controlled SPIMs and proposes a vector control strategy for a symmetrical two-phase induction motor (TPIM) drive (as vector-controlled TPIM) as a viable replacement for the vector-controlled SPIM. The implementation of the vector-controlled TPIM is simpler and more accurate than a vector-controlled SPIM because

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inductances and the rotor self-inductances in the d-q axis, respectively. Llds , Llq s and Lldr , Llq r are the leakage stator inductances and the leakage rotor inductance in the d-q axis, respectively. Ordinarily the value of Ldr is similar to the value of Lq r , therefore the Ldr and Lq r are expressed as Lr (=Ldr = Lq r ) [13]. The constant a is defined as the winding turn ratio and is given by the following: a2 =

Ldm Lds ∼ n2ds = = 2 . Lq m Lq s nq s

(5)

The instantaneous electromagnetic torque Te can be expressed in terms of the flux linkages and currents by [13] P (Lq s isq s isdr − Lds isds isq r ). (6) 2 Te and the load torque TL are related by the following equation: P dωr (Te − TL ) = J + F ωr (7) 2 dt where P, J, and F are the number of machine poles, inertia, and viscous friction coefficient, respectively. Te =

Fig. 2. frame.

Equivalent circuit of an unsymmetrical TPIM in a stationary reference

symmetrical TPIM are used instead of unsymmetrical SPIM. The vector control strategy for a TPIM drive is derived from the vector control strategies used in three-phase ac motor drives [11], [12]. The validity of the proposed symmetrical TPIM vector control strategy was verified by simulations and experiments. II. CONVENTIONAL VECTOR-CONTROLLED SPIM MODEL A. SPIM Model SPIMs are classified as unsymmetrical TPIMs. Fig. 2 shows the dq-axis equivalent circuit of an unsymmetrical TPIM in terms of the stationary reference frame [13]. This equivalent circuit is more complicated than that of three-phase induction motors because the auxiliary winding has more turns than the main winding. The dynamic SPIM model neglecting core saturation and iron losses can be described using a stationary reference frame [13] s vds

vqss

=

rds isds

=

rq s isq s

+

pλsds ,

λsds

+

pλsq s ,

λsq s

=

Lds isds

=

Lq s isq s

0 = rr isdr + pλsdr + ωr λsq r , 0=

rr isq r

+

pλsq r

−

ωr λsdr ,

+

Ldm isdr

(1)

+

Lq m isq r

(2)

λsdr = Lr isdr + Ldm isds

(3)

λsq r

(4)

=

Lr isq r

+

Lq m isq s

where p is the derivation operator and ωr is the rotor angular s , vqss , and isds , isq s are the stator voltages and the staspeed. vds tor currents in the d-q axis in terms of the stationary reference frame(superscript s). λsds and λsq s are the fluxes in the dq-axis of the stator. λsdr and λsq r are the fluxes in the dq-axis of the rotor. rds , rq s , and rr are the stator resistances in the dq-axis and rotor resistance, respectively. Lds (= Llds + Ldm ), Lq s (= Llq s + Lq m ), and Lr are the stator self-inductances in the dq-axis and the rotor self-inductance, respectively. Llds , Llq s , and Llr are the leakage stator inductances in the dq-axis and leakage rotor inductance, respectively. Ldm and Lq m are the magnetizing inductances in the d-q axis. Lds (=Llds + Ldm ), Lq s (=Llq s + Lq m ) and Ldr (=Lldr + Lm r ), Lq r (=Llq r + Lm r ) are the stator self-

B. Modified SPIM Model Many studies have concentrated on modified SPIM models to eliminate the unbalanced operation. This section examines one of the SPIM models, which was presented by Correa [6]. The mutual inductances are not identical in the torque (6). The ac term of the electromagnetic torque can be eliminated by adjusting the stator currents. The phase currents, stator voltages, and counterelectromotive forces in the dq-axis must be redefined to characterize the modified SPIM models due to the adjustment of a isds = isds1 ,

isq s = aisq s1

(8)

esds = esds1 ,

esq s = esq s1 /a.

(9)

Substituting (8) into (6), the torque equation is reexpressed as follows:  P Ldm  s s iq s1 λdr − isds1 λsq r . (10) Te = 2 Lr This equation shows that the symmetrical machine does not produce any torque oscillation in the steady state. From (3) and (4), the rotor-flux model in terms of the stationary reference frame can be expressed as λs dλsdr 1 = − dr − ωr λsq r + Ldm isds1 dt τr τr

(11)

dλsq r λs 1 = − dr + ωr λsdr + Lq m isq s1 dt τr τr

(12)

with τr = Lr /rr . The rotor-flux model in terms of the synchronous reference frame is given by λe dλedr 1 = − dr + ωsl λeq r + Ldm ieds1 dt τr τr

(13)

dλeq r λe 1 = − dr − ωsl λedr + Ldm ieq s1 dt τr τr

(14)

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where ωsl (= ωe − ωr ) is the slip angular speed and ωe is the synchronous angular speed. The currents supplied to the machine need to be oriented in phase and in quadrature to the rotor flux. This can be accomplished by choosing ωe so that the rotor flux is entirely in the d-axis, resulting in λeq r = 0,

λedr = constant.

(15)

This expresses the field orientation concept in the dq variables. Therefore, (10) is reduced to Te =

P Ldm e e λ i . 2 Lr dr q s1

(16)

From (13) to (16), the slip angular frequency becomes Ldm ieq s1 . ωe − ωr = ωsl = τr λedr

(17)

The rotor position angle θe can be obtained indirectly by summing the rotor position angle θr and slip position angle θsl . The SPIM models to compensate for the unbalanced operation in terms of the stationary reference frame can be derived from (1)–(4) and are given by the following:   L2 s = rds + dm isds + σds Lds pisds + esds (18) vds τr Lr   L2q m s vq s = rq s + isq s + σq s Lq s pisq s + esq s (19) τr Lr where

  Ldm λs ωr λsq r + dr Lr τr  s  λq r Lq m = ωr λsdr − Lr τr

esds = −

(20)

esq s

(21)

where σds (= 1 − L2dm /Lr Lds ) and σq s (= 1 − L2q m /Lr Lq s ) are the leakage factors in the dq-axis. esds and esq s are the counterelectromotive forces in the dq-axis. The compensated stator voltage equations in terms of the synchronous reference frame can be rewritten considering (8) and (9) as [6]   die L2dm e ieds1 + σds Lds ds1 + eeds1 + ceds1 vds1 = rds + τr Lr dt (22)   e 2 diq s1 Lq m + eeq s1 + ceq s1 vqes1 = rq s + ieds1 + σq s Lq s τr Lr dt (23) where the modified counterelectromotive forces are given by Ldm λr − ωe σds Lds ieq s1 (τr Lr )

(24)

Ldm ωr λr + ωe σds Lds ieds1 Lr

(25)

eeds1 = − eeq s1 =

where the compensated electromotive forces are given by ceds1 = −(a2 Lq s − Lds )ieq s1

(26)

ceq s1 = (a2 rq s − rds )ieq s1 .

(27)

Fig. 3. Implementation of the vector-controlled SPIM drive using the indirect field orientation.

Fig. 3 illustrates the implementation of the vector-controlled SPIM using the indirect field orientation. The proportionalintegral controllers are represented by a symbol; ωr∗ and λ∗dr represent the reference rotor speed and reference rotor flux of the d-axis, respectively. Correa insists that the SPIM model indicated by (22) and (23) is almost symmetrical when the machine undergoes an unbalanced operation only due to the turn ratio of the two windings. On the other hand, his opinion has mathematical problems. III. PROBLEMS IN VECTOR-CONTROLLED SPIM DRIVES AND ITS MANUFACTURING COSTS The capacitor motor is ordinarily used in the implementation of vector-controlled SPIMs because of their availability. The SPIM has different winding turns and wire diameters in their main windings and auxiliary windings. Therefore, electromagnetic theory states that the primary resistances of the main windings and auxiliary windings are proportional to the winding turns, and are inversely proportional to their wire area [14]. This means that the parameters of the two windings are not identical causing several problems in SPIMs used as unsymmetrical TPIMs. A. Problems Found in SPIM Model Equations The SPIM model equations from (1)–(4), which apply to the unsymmetrical ac machine, are defined by employing a stationary reference frame. The mathematical problems regarding the conventional vector-controlled SPIM are analyzed in this section. The SPIM model can be analyzed classically by revolvingfield theory [1]. Fig. 4 shows the equivalent circuit of the main winding (or d-axis) when an unbalanced stator voltage is supplied to the SPIM. The two stator voltages are composed of the forward voltage Vf and backward voltage Vb . Fig. 5 shows the relationship between the main winding voltage Vm and auxiliary winding voltage Va through the use of a phasor diagram. The forward voltage component of two winding voltages is balanced: aVbf = jVm f (= jVf ), considering the winding turn ratio. The same is true for the backward voltages: aVbb = −jVm b (= jVb ) [1]. The two revolving fields rotate in opposite directions to

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Vb r1 + jx1 + Zb

(33)

−xm x2 + r2 /s + j(xm + x2 )

(34)

Ib = where Zf = Zb =

x2 /s

−xm x2 + r2 /(2 − s) x2 /(2 − s) + j(xm + x2 )

(35)

where s is the slip (= (ωe − ωr )/ωe ). The same is true for the main winding current Im and auxiliary winding current Ia . Im and Ia are composed of the forward current If and backward current Ib , respectively, as shown in Fig. 5 Fig. 4.

Equivalent main winding circuit for SPIM under unbalanced operation.

Im = If + Ib

(36)

Ia = jaIf − jaIb .

(37)

Therefore, If and Ib can be expressed as follows:   1 Ia If = Im − j 2 a   1 Ia Im + j . Ib = 2 a

(38) (39)

The torque generated by each component can be evaluated easily, and it is expressed as [1]  r2 r2 2 2 If2 2 − Ib2 (40) To = ωo s (2 − s)

Fig. 5. Phasor diagram of two unbalanced stator currents and two unbalanced stator voltages.

each other. Therefore, Vm and Va are expressed in terms of Vf and Vb Vm = Vf + Vb Va = j

Vb Vf −j . a a

(28) (29)

Vf =

1 (Vm − jaVa ) 2

(30)

Vb =

1 (Vm + jaVa ). 2

(31)

The forward current If and backward current Ib can be determined using the forward and backward impedances, respectively, in the equivalent circuit in Fig. 4 Vf r1 + jx1 + Zf

where Af (a, φ) =

2 + (Ia /a)2 + 2Im (Ia /a) sin φ} {Im 4

(42)

Ab (a, φ) =

2 + (Ia /a)2 − 2Im (Ia /a) sin φ} {Im 4

(43)

x2m {(r2 /s)2 + (x2 + xm )2 }

(44)

C(s) = D(s) =

Vf and Vb can be expressed in the following:

If =

where If 2 and Ib 2 represent the rotor currents generated in the forward revolving field and the backward revolving field, respectively. Substituting (38) and (39) into (40) yields the total torque equations  2 r2 r2 To = Af (a, φ)C(s) − Ab (a, φ)D(s) (41) ωo s (2 − s)

(32)

{(r2 /(2 −

x2m . + (x2 + xm )2 }

s)2

(45)

The first term in (41) defines the positive torque generated by the forward revolving field, and the second term defines the negative torque generated by the backward revolving field. The total torque is changed by adjusting the turn ratio a, the phase difference angle Φ, and the slip S. The negative torque in the second term of (43) can be eliminated if the magnitude of Im is identical to that of (Ia /a), and the phase difference angle Φ between the two currents is fixed at 90◦ . On the other hand, it is difficult to fix the phase difference angle at 90◦ and maintain two currents with the same magnitude because the impedances of the two windings in (34) and (35) vary according to the slip.

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Fig. 6.

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Simulation results due to S. (a) Magnitude of Im and Ia . (b) Phase difference angle.

Fig. 6 shows the simulation results for the magnitude of Im and Ia and the phase difference angle Φ between the two phase currents when the SPIM speed changes from 1000 to 1780 r/min with the following test motor parameters: Vm = 220 V, Va = 220 V, a( = Nm /Na ) = 1.3, r1m = 0.416 Ω, L1m = 42.4 mH, Lm = 42.4 mH, L2m = 35.6mH, r2m = 0.646 Ω, and r1a = 0.956 Ω. Fig. 6(a) shows that the magnitude of the two phase currents deceases with increasing motor speed. In addition, the magnitude of Im is higher than Ia due to the low r1m . Fig. 6(b) shows that the phase difference angle Φ increases with increasing motor speed to a maximum value and then decreases rapidly with further increasing motor speed. Therefore, it is difficult to eliminate the negative torque from (41). B. Complex Implementation of the Vector-Controlled SPIM Drive System Conventional vector-controlled SPIM methods have concentrated mostly on eliminating unbalanced operation because vector control strategies are based on a balanced drive system involving symmetrical motors. Modified stator voltage equations are needed to compensate for the unbalanced SPIM operation, e.g., ceds1 and ceq s1 in (22) and (23). Therefore, the implementation of a vector-controlled SPIM drive is more complex than that a vector-controlled symmetrical motor. Fig. 3 shows the implementation of the vector-controlled SPIM used to accomplish (22) and (23). Therefore, the implementation of a vector controlled SPIM drive is more complex than that a vector controlled symmetrical motor, as shown in Fig. 3.

C. Difficulty in Measuring the SPIM Parameters The parameters for ac machines need to be measured precisely to operate a vector-controlled ac machine without error. The parameters for symmetrical machines can be calculated precisely using the no-load test and locked-rotor test. On the other hand, unsymmetrical motors produce negative and positive torque during operation. This makes measuring the parameters more complicated [1]. In addition, modeling, renewing, and optimizing single-phase capacitor motors are difficult compared to symmetrical motors [17], [18]. Therefore, many calculations and iterations of measuring tests are needed to determine the SPIM parameters. In [18], numerous computer calculations were proposed to obtain precise measurements of the SPIM parameters. Therefore, unsymmetrical machines as SPIM is not proper to control the speed motor by using a vector control strategies, and symmetrical motors should be used as a replacement. D. Difficulty in Calculating the Turn Ratio The turn ratio a from (5) is ordinarily defined as the ratio of the main winding turns to the auxiliary winding turns (= (Nds /Nq s ). But a is precisely defined as the ratio of the effective main winding turns to the effective auxiliary winding turns (= kw d Nds /kw q Nq s ), where kw q (= kpq kdq ) and kw a (= kpa kda ) are the winding factors (= kw d ) multiplied by the pitch factor (= kpq ) and distribution factor (= kw a ) [1]. The winding factor should be considered when calculating the turn ratio a. Although the difference in winding factor (= kw m − kw a ) between the two winding factors is small between the two windings, it affects the value of a. If the value of a in (22) and (23) is not exact, the implementation of the vector-controlled SPIM shown in Fig. 3 cannot maintain balanced operation.

JANG: PROBLEMS INCURRED IN A VECTOR-CONTROLLED SINGLE-PHASE INDUCTION MOTOR

Fig. 8.

531

Implementation of a indirect vector-controlled TPIM.

ple, the five-phase induction motor and synchronous reluctance motor drives, which have not been manufactured in any quantity until now, have been studied for future use owing to their advantages. IV. VECTOR CONTROL STRATEGY FOR TPIM DRIVE This section proposes a vector control strategy for symmetrical TPIMs (as “ vector-controlled TPIM “) as a replacement for the vector-controlled SPIM. Historically, the vector control strategies have concentrated on three-phase ac machines and have not been attempted in symmetrical TPIM until now. The vector-controlled TPIM can solve several of the problems that plague the vector-controlled SPIM, and can control the speed of the TPIM precisely. Fig. 7.

Four space vectors in the SPIM drive. (a) k < 1. (b) k >1.

Fig. 7 shows four space vectors in a two-leg inverter and attainable voltage range in the complex plane. In the figure, each “1” represents an output line attached to the positive dc link, whereas “0” represents an output line to the negative dc link. The four space vectors are distributed evenly at 90◦ intervals with a length of Vdc /2 and form an exact square [15], [20]. Assuming that the value of a is exact in the implementation of the vectorcontrolled SPIM, the achievable voltage range decreases in the complex plane due to (9), when the value of a is higher or lower than 1. On the other hand, when a = 1, the SPIM is theoretically identical to a symmetrical TPIM, so the attainable voltage range has a maximum value. Although the characteristics of the SPIM are poor, vector control strategies for SPIM have been attempted because SPIMs are used widely and are inexpensive. Researchers, however, need to make a special order to acquire TPIMs for their experiments because the TPIMs are not manufactured. Therefore, the TPIM cost is higher than SPIM due to customization, but it is not overly high because a TPIM can be constructed using the same facilities as an SPIM or a three-phase induction motor. If TPIMs become popular, the real cost of a TPIM will be lower than that of an SPIM because the TPIM does not require a capacitor. On the other hand, special motor drives, for exam-

A. Transformation of Phase Current Equations The phase current equations in terms of the stationary reference frame must be transformed to that in terms of the synchronous reference frame in the vector control, as shown in Fig. 8  s  e  sin θe cos θe id id = . (46) e iq − sin θe cos θe isq B. Symmetrical TPIM Model Symmetrical TPIMs have two identical stator windings that are arranged in an electrical quadrature. Since the rotor windings are forged as a squirrel cage, the parameters of dq-axis are identical to each other and are given by the following equation: Lds = Lq s = Ls Ldr = Lq r = Lr Ldm = Lq m = Lm (47) rds = rq s = rs rdr = rq r = rr . These parameters transform an unsymmetrical TPIM to a symmetrical one. The symmetrical TPIM model in terms of the synchronous reference frame can be rearranged using (1)–(4) and (47) e = rs ieds + pλeds − ωe λeq s , vds

vqes

=

rs ieq s

+

pλeq s

+

ωe λeds ,

λeds = Ls ieds + Lm iedr

(48)

λeq s

(49)

=

Ls ieq s

+

Lm ieq r

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0 = rr iedr + pλedr − ωsl λeq r ,

λedr = Lr iedr + Lm ieds (50)

0 = rr ieq r + pλeq r + ωsl λedr ,

λeq r = Lr ieq r + Lm ieq s (51)

where ω sl (= ω e −ω r ) is the slip angular speed and ω e is the synchronous angular speed. The torque equation Te in terms of synchronous reference frame can be expressed as Te =

P Ldr e e (i λ − ieds λeq r ). 2 Lr q s dr

(52)

C. Indirect Vector Control for Symmetrical TPIM Vector control strategies have been developed focusing on three-phase ac motors [11]. On the other hand, the vectorcontrolled TPIM as a replacement for vector-controlled SPIM has not been proposed until now, even though it is similar to the vector-controlled three-phase ac motors. This paper proposes a simple indirect vector control strategy for TPIM, which can be applied to the low-power motor drive system. The currents supplied to the machine need to be oriented in phase and in quadrature for the rotor flux. This can be accomplished by choosing ωe with the rotor flux entirely in the d-axis, resulting in the following: λeq r = 0,

λedr = constant.

(53)

The torque (52) of the TPIM is reduced to Te =

P Lm e e λ i . 2 Lr dr q s

Lm ieq s . τr λedr

(55)

The voltage equations of the TPIM model in terms of the synchronous reference frame can be expressed as     Lm e Lm e e e e pλ λ vds = (rs + Lx p) ids + − ωe Lx iq s + Lr dr Lr q r

vqes

e = vds + eeds (56)     Lm e Lm e = (rs + Lx p)ieq s + pλq r + ωe Lx ieds + λdr Lr Lr

= vqes + eeq s

(57)

Implementation of a indirect vector-controlled three-phase ac motor.

D. Differences Between the Vector-Controlled TPIM and the Vector-Controlled Three-Phase AC Motor The vector control strategy for symmetrical TPIM is derived from the vector-controlled three-phase ac machine. Three-phase ac motors are operated by a rotating mmf. The rotating mmf vector Fss is determined by the addition of the stationary threephase mmfs vectors Fss = Fas + aFbs + a2 Fcs = Fds + jFqs

(54)

The slip angular frequency yields the following equation, which is similar to (16) in the modified SPIM model: ωe − ωr = ωsl =

Fig. 9.

(59)

where a = ej 2π /3 = cos 23 π + j sin 23 π. The abc → dq transformation for the phase currents in terms of the stationary reference frame is needed for vector-controlled three-phase ac motors

 √ 3 2 1 ias − (ibs + ics ) + j (ibs − ics ) = isd + jisq . = 3 2 2 (60) Therefore, the matrix equation becomes

isab c



isd isq

⎡

=

2⎢ ⎣ 3

1 0

1 2 √ 3 − 2

−

1 ⎤⎡i ⎤ as 2 ⎥⎢ √ ⎦ ⎣ ibs ⎥ ⎦. 3 − ics 2 −

(61)

The abc → dq transformation and inverse transformation are omitted in the implementation of the indirect vector-controlled (58) TPIM, as shown in Fig. 8, whereas transformations are needed in where Lx is the stator transient inductance and σ is the leakage the indirect vector-controlled three-phase ac machine, as shown parameter. eeds and eeq s are defined as the counterelectromotive in Fig. 9. Two output currents in a TPIM drive system can be force. The voltage equations become simpler because the orien- measured using only a single current sensor [19], whereas the tation force λeq r is zero and λedr is constant. three output currents can be measured using two current sensors Fig. 8 shows the implementation of the indirect vector- in a three-phase ac motor drive system. If the measured isds value controlled TPIM drive. The field angle θe is measured by sum- in the d-axis is Im cos θe by one current sensor, it is supposed ming the rotor position signal and slip position signal by cal- that isq s value in the q-axis is Im sin θe . The arrangement of two stator windings in the slots of TPIM culating (55). The turn ratio a and compensated electromotive forces ceds and ceq s in (22) and (23) are omitted, and eeds and eeq s is simpler compared to three-phase ac machines. Therefore, the are simpler in the vector-controlled TPIM, whereas ceds and ceq s TPIM drive system is useful in low-power motor drive applicaare needed and eeds and eeq s are complex in the vector-controlled tions and in areas where only a single-phase voltage source is SPIM. available. L2 Lx = Ls − m = σLs Lr

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Fig. 10.

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Determination of the switching period in a two-leg inverter. Fig. 11. Two reference voltages, triangular carrier wave, and two output voltages in a two-leg inverter.

V. SINUSOIDAL PULSEWIDTH MODULATION (PWM) TECHNIQUE FOR THE TWO-LEG INVERTER Although many PWM techniques for voltage source inverters have been proposed, the sinusoidal PWM technique is the simplest method for an inverter [20]. In this paper, the sinusoidal PWM technique was used for the two-leg inverter-fed TPIM, as shown in Fig. 1(b). The offset voltage specified by Vsn exists between the phase voltages Vds and Vq s across the loads, and the output voltages Vdn and Vq n exist in the two-leg inverter. Therefore, the determined output voltages Vdn and Vq n can be expressed as follows: Vdn = Vds + Vsn

(62)

Vq n = Vq s + Vsn .

(63)

TABLE I PARAMETERS OF THE TPIM MODEL USED IN THE SIMULATION AND EXPERIMENT

The output voltages and the phase voltages are identical because Vsn is zero in Fig. 1(b). The switching times of each leg are determined by a comparison between the Vdn , Vq n , and the triangular carrier wave, as shown in Fig. 10. The on times T1 and T2 of the upper switches used to realize Vdn and Vq n can be expressed as [20] T1 =

Vdn Ts + Ts 2 Vdc

(64)

T2 =

Vq n Ts + Ts . 2 Vdc

(65)

Fig. 11 shows the simulated waveforms for the two reference voltages, triangular carrier wave, and two output voltages in the two-leg inverter when the frequency of the reference voltage is fixed at 60 Hz. The sampling time is approximately 270 μS and the MI is fixed to 0.8. The output voltages of the two-level PWM waveform [−200 V, 200 V] are constructed by the intersecting points between the carrier wave and two reference voltages. VI. SIMULATION RESULTS The proposed strategy was verified by simulations using MATLAB. Table I lists the parameters of the TPIM model (rated voltage: 220 V, rated current: 5 A, rated power: 1.4 kW, rated

Fig. 12. Transient response for the motor speed, reference q-axis stator current, response for the q-axis stator current, and rotating q-axis stator current due to the variation of speed under 30% load torque (400 r/min ⇒ 800 r/min).

speed: 1760 r/min, and the number of poles: 4), which can be calculated using the no-load test and locked-rotor test. Fig. 12 shows the transient response for the motor speed, reference q-axis stator current Iqe∗s , its response Iqes for the qaxis stator current, and the rotating q-axis stator current Iqss when the reference speed is changed suddenly from 400 to 800 r/min under a 30% load torque. The q-axis stator current Iqes is in proportion to the load torque except during the transient

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Fig. 13. Transient response for the motor speed, reference q-axis stator current, response for the q-axis stator current, and rotating q-axis stator current under a fixed 800 r/min due to a variation of the load (no load ⇒30% load ⇒ no load).

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Fig. 15. Transient response for the motor speed and two rotating stator currents in the dq-axis under no load due to the slow variation in the opposite direction of the reference speed (−800 r/min ⇒ 800 r/min).

Fig. 16. Experimental waveforms for the steady-state two-phase currents and their locus.

Fig. 14. Transient response for the motor speed, reference q-axis stator current, and rotating q-axis stator current under no load due to the quick variation in the opposite direction of the reference speed (−400 r/min ⇒ 400 r/min ⇒ −400 r/min).

response period; its magnitude is approximately 2.5 A. The actual frequency of Iqss is in proportion to the reference speed. Fig. 13 shows the transient response for the motor speed, Iqe∗s , Iqes , and Iqss , in the q-axis when the load torque is changed suddenly from no load to a 30% load torque and returned to a no load at a fixed speed of 800 r/min. The motor speed maintains a constant 800 r/min regardless of the magnitude of the load torque. The magnitude of the rotating stator current Iqss increases but its frequency is fixed due to the motor speed. Fig. 14 shows the reference motor speed, transient response for the motor speed, and reference q-axis stator current Iqe∗s , along with its response for Iqe∗s when the reference speed changes suddenly in the opposite direction from −400 to 400 r/min under

no load. The actual motor speed followed the reference motor speed, and the magnitude of Iqe∗s and Iqss was zero due to the no load, and the actual frequency of Iqss maintains a constant value except during the transient response period. Fig. 15 shows the transient response for the motor speed and two rotating stator s in the dq-axis when the reference speed currents Iqss and Ids changes to the opposite direction from −800 to 800 r/min slowly under the no-load condition. The d-axis rotating stator current s leads the q-axis rotating stator current Iqss by 90◦ , and after Ids s lags Iqss by 90◦ . This shows that two the transient period, Ids phase currents with a phase difference angle of 90◦ are balanced in the dq-axis. The simulation shows that the performance of the vector-controlled TPIM is similar to those observed for the three-phase induction motor. VII. EXPERIMENTAL RESULTS The motor parameters of the TPIM model used for the experiments are the same as those used for the simulation, as stated

JANG: PROBLEMS INCURRED IN A VECTOR-CONTROLLED SINGLE-PHASE INDUCTION MOTOR

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Fig. 17 Experimental response for the motor speed and two q-axis stator currents due to the variation of speed under a 30% load (400 r/min ⇒ 800 r/min).

Fig. 19. Experimental response for the motor speed and two q-axis currents due to the opposite direction of the reference speed under no load (−400 r/min ⇒ 400 r/min ⇒ −400 r/min).

Fig. 18. Experimental response for the motor speed and q-axis stator currents due to variation of load torque at a fixed 800 r/min (no load ⇒ 30% load ⇒ no load).

Fig. 20. Experimental response for the motor speed and dq-axis currents due to the opposite direction of the reference speed under no load (−800 r/min ⇒ 800 r/min).

in Table I. The vector-controlled TPIM was implemented using a TMS320C28335 DSP as the controller. Fig. 16 shows the experiment waveform for the two-phase currents and their locus. The magnitude of the two phase currents was identical and the phase difference angle was 90◦ under a steady state. These results are similar to the simulation waveforms shown in Fig. 15. Fig. 17 shows the experimental response for the motor speed, reference q-axis stator current Iqe∗s , and responses Iqes for Iqe∗s , as well as the rotating q-axis stator current Iqss when the reference speed changes from 400 to 800 r/min under a 30% torque load. The experiment results in Fig. 17 are similar to the simulation waveforms shown in Fig. 12. Fig. 18 shows the experimental response for the motor speed, Iqe∗s , Iqes , and Iqss , when the load torque changes suddenly from no load to a 30% load and returns to no load at a fixed 800 r/min, which are the same conditions shown in Fig. 13. The experiment waveforms in Fig. 18 are similar to the simulation waveforms shown in Fig. 13. Fig. 19 shows the experimental response for the motor speed, Iqes and Iqss , when the reference speed changes suddenly to the opposite direction from −400 to 400 r/min and then to the opposite direction from 400 to −400 r/min under no load. The experiment results for the motor speed, Iqes and Iqss , were similar to the simulation waveforms shown in Fig. 14. Fig. 20 shows the experimental response for the motor speed and the two rotating stator currents Iqss and

s Ids in the dq-axis when the reference speed changes slowly to the opposite direction from −800 to 800 r/min. The experiment waveforms were similar to the simulation waveforms shown in Fig. 15. The motor parameters of the TPIM model used for the experiments are the same as those used for the simulation, as stated in Table I. The vector-controlled TPIM was implemented using a TMS320C28335 DSP as the controller.

VIII. CONCLUSION This paper presented the problems encountered in the conventional vector-controlled SPIM, and explained that the vector control strategies for the SPIM drive are not needed in the low power level field. The vector-controlled TPIM was presented as a replacement for the vector-controlled SPIM. The proposed vector-controlled TPIM has several advantages. The implementation of the proposed vector-controlled TPIM is simpler than that of the vector-controlled SPIM. The parameters of the TPIM can be calculated simply, whereas parameter measurements for an SPIM are difficult. The vector controlled TPIM is useful to the low power motor drive applications at area where the voltage source in the drive system can be supplied by the dc battery or single-phase ac voltage source. The vector-controlled TPIM was derived from the indirect vector control concept for three-phase ac machines. In addition,

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several differences between the vector control strategies for the TPIM and three-phase ac motors were explained. The proposed vector control strategy is useful for low-power motor drives and in areas where a single-phase voltage source is available. The validity of the vector control strategy for the symmetrical TPIM was verified through simulations and experiments. REFERENCES [1] Cyril G. Veinott, Theory of Design of Small Induction Motors. New York: McGraw-Hill, 1959. [2] F. Blaabjerg, F. Lungeanu, K. Skaug, and M. Tonnes, “Two-phase induction motor drives,” IEEE Ind. Appl. Mag., vol. 10, no. 4, pp. 24–32, Jul./Aug. 2004. [3] C. Mdemlis, I. Kioskeridis, and T. Thodoulidis, “Optimization of singlephase induction motors—Part I: Maximum energy efficiency control,” IEEE Trans. Energy Convers., vol. 20, no. 1, pp. 187–195, Mar. 2005. [4] M. F. Rahman and L. Zhong, “A current-forced reversible rectifier fed single-phase variable speed induction motor drive,” in Proc. 27th Annu. IEEE Power Electron. Spec. Conf., Jun., 1996, pp. 114–119. [5] M. B. R. Correa, C. B. Jacobina, A. M. N. Lima, and E. R. C. da Silva, “Field oriented control of a single-phase induction motor drive,” in Proc. IEEE Power Electron. Spec. Conf., May 1998, pp. 990–996. [6] M. B. R. Correa, C. B. Jacobina, A. M. N. Lima, and E. R. C. da Silva, “Rotor-flux-oriented control of a single-phase induction motor drive,” IEEE Trans. Ind. Electron., vol. 47, no. 4, pp. 832–841, Aug. 2000. [7] M. B. R. Correa, C. B. Jacobina, and E. R. C. da Silva, “Vector control strategies for single-induction motor drive system,” IEEE Trans. Ind. Electron., vol. 51, no. 5, pp. 1051–1080, Oct. 2004. [8] S. V. Zadeh and S. Harooni, “Decoupling vector control of single phase induction motor drives,” in Proc. IEEE Power Electron. Spec. Conf., 2005, pp. 733–738. [9] D. Wang, “Hybrid fuzzy vector control for single phase induction motor,” in Proc. Int. Conf. Comput. Control Ind. Eng., 2010, pp. 122–125. [10] A. Nied, J. Oliveira, F. R. L. Sa, R. Campos, and L. H. R. C. Stival, “Singlephase induction motor indirect field oriented control under nominal load,” in Proc. Power Electron. Drive Syst., 2010, pp. 789–793. [11] A. M. Trzynadlowsky, The Field Orientation Principle in Control of Induction Motors. Norwell, MA: Kluwer, 1994.

[12] V. T. Phan, H. H. Lee, and T. W. Chun, “An improved control strategy using a PI resonant controller for an unbalanced stand-alone doubly-fed induction generator,” J. Power Electron., vol. 10, no. 2, pp. 194–202, 2011. [13] Paul C. Krause, Analysis of Electric Machinery. New York: McGrawHill, 1987. [14] G. R. Slemon and A. Straughen, Electric Machinery. Reading, MA: Addison-Wesley, 1980. [15] D.-H. Jang and D.-Y. Yoon, “Space vector PWM technique for two-phase inverter-fed two-phase induction motors,” IEEE Trans. Ind. Appl., vol. 39, no. 2, pp. 542–549, Mar./Apr. 2003. [16] A. M. A. Jabbar, A. M. Khambadkone, and Z. Yangfeng, “Space-vector modulation in a two-phase induction motor drive for constant-power operation,” IEEE Trans. Ind. Electron., vol. 51, no. 5, pp. 1081–1088, Oct. 2004. [17] C. Attaianese, A. Del Pizzo, E. Pagano, and A. Perfetto, “Modelling, renewing and optimizing single-phase capacitor motors for home applications,” in Proc. IEEE Ind. Appl. Soc. Conf. Rec, 1986, pp. 831–839. [18] E. F. Fuchs, A. J. Vandenput, J. Holl, and J. C. White, “Design analysis of capacitor-run single-phase induction motors,” IEEE Trans. Energy Convers., vol. 5, no. 2, pp. 327–336, Jun. 1990. [19] Y. H. Cho, K. Y. Cho, H. S. Mok, K. H. Kim, and J. S. Lai, “Phase current reconstruction techniques for two-phase inverters using a single current sensor,” J. Power Electron., vol. 11, no. 6, pp. 837–845, 2011. [20] D.-H. Jang, “PWM methods for two-phase inverters,” IEEE Ind. Appl. Mag., vol. 13, no. 2, pp. 50–61, Mar./Apr. 2007.

Do-Hyun Jang (S’87–M’02) received the B.S. degree in electrical engineering from Hanyang University, Seoul, Korea, in 1980, and the M.S. and Ph.D. degrees from Seoul National University, in 1982 and 1989, respectively. Since 1985, he has been with the Department of Electrical Engineering, Hoseo University, Chungnam, Korea, where he is currently a Professor. From 1993 to 1994, he was a Visiting Scholar in the Department of Electrical Engineering, Texas A&M University, College Station. His research interests include ac motor speed control, switched reluctance motor drives, and two-phase induction motor drives.