Electric Power Systems Research 73 (2005) 121–128

Three-phase induction motor operating from single-phase supply with an electronically controlled capacitor Nabil A. Ahmed Department of Electrical and Electronics Engineering, Assiut University, Assiut 71516, Egypt Received 27 October 2003; received in revised form 26 May 2004; accepted 20 June 2004 Available online 25 September 2004

Abstract In this paper the performance of a three-phase induction motor operating from single-phase supply with a new electronically controlled capacitor using an electronic switch in series with a fixed capacitor to achieve a minimum unbalance of the motor phase voltages at all loading conditions is proposed. No mechanical or centrifugal switch is used here. Basic system operation, theoretical analysis, simulated and experimental results in comparison with conventional operation using one and two fixed capacitors are presented in this paper. The optimum effective capacitor value can be on-line adjusted at any operating speed by periodically changing the duty cycle of the controlled switch to achieve minimum unbalance in phase voltages to improve the motor performance at different speeds. © 2004 Elsevier B.V. All rights reserved. Keywords: Controlled capacitor; Duty cycle; Induction motor; Minimum unbalance

1. Introduction The three-phase supply system in now available worldwide, except perhaps in some rural areas where only a singlephase supply is available. Single-phase motors are the most common form in the lower horse-power ranges, but they become uneconomical for ratings above about 0.5 kW and therefore an increasing tendency to use standard three-phase motors supplied from single-phase supply if the three-phase supply is not available [1–3]. In many applications it may be become necessary to use three-phase induction motor on a single-phase supply system. For example, it has been found technically and economically advantageous to install initially a single wire earth return (SWER) system for rural electrification in remote and hilly regions. As the load demand increases, the SWER system can be converted to regular threephase power system. Irrigation pump sets, which often form the major load on the system, may use either a single-phase or a three-phase motor with a capacitor phase converter, the latter having the advantages that it can be used directly when E-mail address: [email protected]. 0378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2004.06.007

the three-phase supply is installed later. Another example is the portable pump sets which must operate with three- or single-phase supply depending on what is available at the site [4]. It is well known that when an three-phase induction motor is connected to a single-phase supply system, no starting torque is developed unless the motor has some form of a poly-phase primary winding in which poly-phase voltages are induced. This form can be achieved by connecting a phase capacitor in series with one of the motor phases. The function of the capacitor is to generate a leading phase so that the motor operates as a near balanced two-phase machine. For this reason, the capacitor value must be carefully chosen according to the motor impedance. Unfortunately, this impedance changes dramatically from starting to running. The problems incurred in this aspect are essentially those of selecting the value of the capacitor to give satisfactory starting and running performance. Many papers have proposed different control techniques to improve the performance of three-phase induction motor operating from single-phase supply and well documented in the literature [3–9]. These papers conclude that the overall

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rating of a three-phase motor operating with single-phase supply, with proper choice of the capacitor phase converter, could be as high as 70% of a balanced rating three-phase motor for the same frame size. This makes the three-phase motor almost as good as a well designed single-phase motor. Further, the three-phase motor is cheaper than single-phase motor. There are many different criteria are used to optimize the capacitor size such as minimum unbalance of the phase voltages, power factor, input power, efficiency, torque or the input current to give satisfactory operation of the motor [4,7,9,14]. This paper uses the criteria of minimum unbalance of the phase voltages at all motor speeds. It can be shown that ideally the capacitor should be varied continuously with the speed of the machine, but in practice the choice is limited to two set capacitor values, as in the case of split-phase motors, one is used for starting period to increase the starting torque and the other is used during running condition at normal speed to improve the efficiency and the motor performance [3–5], the change-over being effected manually by means of a centrifugal switch; the most common failure is that the switch will ultimately fail to open when needed. Moreover, if the speed of the motor is changed for any reason such as the applications required variable speed operation or even due to any change in the loading conditions, the value of the fixed running capacitor becomes not suitable and another value of the running capacitor must be selected. To the author’s knowledge, continuously adjusting the capacitor size with the speed of three-phase induction motors connected to single-phase supplies is not yet discussed. A controlled capacitor has been proposed and used to improve the performance of split-phase induction motors by Lipo et al. [10–13]. The system uses a dc charged capacitor switched by a bidirectional switch represented by a transistor H-bridge. The switch is short-circuited and opened for each cycle; low switching frequency. Also, the switch is closed at the instant the voltage across the capacitor reaches zero; thus a zero voltage switching must be performed. The object of this paper is to introduce an alternative configuration of the controlled capacitor by using a fixed capacitor in series with an electronic switch and to supply a continuously variable capacitance to three-phase induction motors operating from single-phase supplies. No mechanical or centrifugal switch is used, high switching frequency can be used and no zero voltage switching must be performed. By suitably adjusting the duty cycle of the electronic switch, the required value of the capacitor can be achieved according to the desired motor performance. The other distinctive feature in this paper is the implementation of an on line adjustment of the controlled capacitor and the validation of the developed mathematical model by computer simulation and experimental work. The system consists of three major elements: an induction motor, a switch controlled capacitor and a control circuit. A large capacitor required for starting is permanently connected in series with one of the motor phases and a single

unidirectional switch is used with the aid of a diode bridge rectifier to constitute a bidirectional switch representation. It is quite clear that the scheme is simple as it composes of only one controlled switch which is essential for reliability and effective cost. The paper is organized as follows: a mathematical model for three-phase induction motor operating with single-phase supply is given in Section 2. This model is established by using instantaneous symmetrical component and a steady state equivalent circuit is derived. The principle of a controlled capacitor is proposed in Section 3 and then the duty cycle of the controlled capacitor to obtain minimum unbalance of the phase voltages can be determined. Section 4 provides both simulated and experimental results. Finally concluding remarks are given in Section 5.

2. Mathematical model Since the stator windings of the three-phase induction motor operating with single-phase supply are not identical, it is necessary to use a transformation technique in order to obtain the physical variables. The instantaneous symmetrical component transformation is used to obtain voltage and current equations as positive and negative components. This technique has been adopted in formulating the dynamic equations of three-phase induction motors operating from single-phase supply [5,14]. In this analysis, all the machine parameters are assumed constant. Core losses, magnetic saturation and mmf space harmonics are neglected. For the stator winding of a star-connected three-phase induction motor connected to a single-phase supply with electronically controlled capacitor represented in Fig. 1, where two winding connected directly to the supply (phases a and b) while the phase c winding is connected to the supply through the capacitor, zero sequence quantities would be absent (three-wire system) and the sum of the three-phase currents is always to be zero. Further, the negative sequence quantities are always the complex conjugates of the corresponding positive sequence quantities. Thus, only the positive sequence quantities are required for complete solution.

Fig. 1. Stator connection of single-phase operation of three-phase induction motor with electronically controlled capacitor.

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The terminal voltage relations for the stator circuit fed from single-phase supply and connected in Fig. 1 are  1 va − iC dt − vc = 0, vs = va − vb , C va −

XC ic − vc = 0 P

(5)

where XC = 1/ωC is the capacitive reactance. While the current relations are ia + ib + ic = 0,

Fig. 2. Positive sequence operational equivalent circuit.

is = ia + ic ,

ib = −is

(6)

Applying the instantaneous symmetrical component analysis Following the instantaneous symmetrical components transformation and from the positive and negative sequence equivalent circuit of three-phase induction motor shown in Fig. 2, the resulting performance equations in terms of sequence quantities and motor parameters will be [14,15]:

1 2 − vs = √ {v+ s (1 − a ) + vs (1 − a)}, 3 1 + XC + 2 0 = √ {vs (1 − a) + v− (ais + a2 i− s (1 − a ) − s )} (7) p 3

+ + + v+ s = (Rs + Xls p)is + Xm p(is + ir ),

By combining Eqs. (3) and (7), the performance Eqs. (1) and (2), can be simplified as √ vs = 3(Rs isx + Xs pisx + Xm pi)

v+ r 1 − j(γ/P)
  Rr  + + = + Xlr p i+ r + Xm p(is + ir ) 1 − j(γ/P)

0=

− (Rs isy + Xs pisy + Xm piry ), (1)













0 = Rr irx + Xr pirx + Xm pisx + γ(Xr iry + Xm isy ),

where γ is the p.u. mechanical speed (ωm ) with the synchronous speed (ωs ) as the base speed, p the operator (1/ω)(d/dt) and ω the supply radian frequency. By the same way the negative sequence operational equivalent circuit is obtained directly by replaced j by −j in (1):

0 = Rr iry + Xr piry + Xm pisy − γ(Xr irx + Xm isx ), √ 0 = 3(Rs isx + Xs pisx + Xm pirx ) + Rs isy + Xs pisy

− − − v− s = (Rs + Xls p)is + Xm p(is + ir ),

where Xs = Xls + Xm , Xr = Xlr + Xm . Substituting from (3) into (4), a simplified expression for the developed electromagnetic torque results as

v+ r 1 + j(γ/P)
  Rr  − − = + Xlr p i− r + Xm p(is + ir ) 1 + j(γ/P)



0=

(2)

i+ s = isx + jisy ,

v+ r = vrx + jvry ,

i+ r = irx + jiry

(3)

where, the corresponding negative sequence terms would just be the complex conjugate. The developed electromagnetic torque in terms of sequence quantities can be expressed as Te =

jXm + − − (i i − i+ s ir ) ωs r s

Te =

(8)



2Xm (irx isy − isx iry ) ωs

(9)

and the mechanical balance torque equation is

Eqs. (1) and (2) are complex. Therefore, it is convenient for simulation to write these equations in terms of their real and imaginary parts. Thus, the positive sequence voltages and currents which are complex numbers can be written in terms of their real and imaginary parts as v+ s = vsx + jvsy ,

1 XC XC + Xm piry + √ isx + isy P 3 P

(4)

dωm + Fωm + TL = Te , P   Te − TL − Fωm pωm = ω J

J

(10)

where F is the friction coefficient. The independent variables of the system can be considered as isx , isy , irx , iry and γ. Eqs. (8)–(10) are all non-linear equations as they involve product of variables, i.e. current with current or current with speed. The numerical techniques must be used to evaluate the system performances at different instants of time starting from the initial values to run-up calculations. The computation was started by defining the motor applied voltage vs and the initial values for all variables were set to zero.

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3. Optimization of the capacitor size

and

From (8) it is easy to understand that the capacitive reactance needed varies with the speed and the parameters of the motor. The proper selection of the capacitor value plays an important role in determining the machine performance. For satisfactory operation of three-phase induction motor with single-phase supply, it is essential to carefully select the size of the motor capacitor C or the value of the capacitor reactance XC such that the effect of negative sequence currents is considerably reduced if not totally eliminated so that the undesirable features such as vibration and noise can be kept at a low level. There are different criteria for selecting the optimum value of XC ; these criteria are minimum unbalance in stator phase windings voltage, minimum negative sequence voltage and zero negative sequence voltage. Here, the criterion of minimum unbalance in the stator winding voltages and zero negative sequence voltage has been used. The value of the capacitor that yields minimum unbalance yields also maximum efficiency [7]. Expressions for the value of the capacitor that can achieve the above requirements can be found in [5,14].

D=

3.1. Principles of the controlled capacitor The switched capacitor circuit shown in Fig. 1 consists of two main parts: a fixed ac capacitor and an electronic switch. The switch must allow bidirectional current flow as in a mechanical switch. The switch is short circuited and opened periodically using symmetrical pulse width modulation (PWM) technique which is simple and easy to implement. No detection of zero crossing of the capacitor voltage is required compared with the switched capacitor proposed in [10–13] where the switch is closed at the instant the voltage across the capacitor reaches zero; thus zero voltage switching is not required. Also, the switch in [10–13] is closed and opened one time for each supply cycle. In the proposed circuit, the switch can be switched many times in each supply cycle and therefore a high switching frequency can be used. Using a high switching frequency for the switched capacitor leads to very short off periods of the switched capacitor. Therefore, the charging periods of the fixed capacitor, while the switched capacitor is off, become very small and consequently a little voltage difference between the capacitors at the instant of switching is performed to avoid any current spikes at the switching instant. By opening and closing the switch periodically, the effective value of the capacitor appears to be smaller than the actual value. Therefore, a controlled capacitor is presented in this section by using a fixed capacitor in series with an electronic switch. A single unidirectional switch is used with the aid of a diode bridge rectifier to constitute a bidirectional switch representation as shown in Fig. 1. The relationship can be approximately expressed as XCeffective = XCreal

T XCreal = Ton D

(11)

Ton T

(12)

where XCreal is the real capacitive reactance, XCeffective is the effective capacitive reactance, T = Ton + Toff is the switching period of the electronic switch, D is the duty cycle, let XC = 1/jωC, and substitute into (11), then we can obtain Ceffective = DCreal

(13)

From Eq. (13) when the switch is closed (D = 1.0), Ceffective is equal to Creal . On other hand, when the shorting interval of the switch is decreased, Ceffective will become small until it equals zero at D = 0 and it is easy to adjust the value of the effective capacitor by changing the shorting interval of the switch. A switched capacitor can change the effective capacitor value from large to small when the speed of the motor is increased and it provides an effective method of overcoming the difficulty associated with the need for a variable capacitor. The hardware circuit consists of three parts: a sawtooth generator, comparator circuit and isolation and driving circuit. The speed of the motor is sensed by a tachometer which aligns with the motor shaft. The electronic switch in series with the capacitor is realized by an IGBT. The switching signal of the switch can be easily obtained by comparing a dc signal having variable amplitude with a high frequency sawtooth signal has a constant amplitude. The dc signal amplitude can be selected as a function of the motor speed to give the required value of the short-circuited and opened periods (duty cycle) of the switch. By suitably adjusting the duty cycle, the required effective value of the capacitor, Fig. 4, can be obtained and minimum unbalanced can be achieved. Also, by adjusting the capacitor size, the backward field would be eliminated or at least minimized with resulting improving in efficiency, torque pulsations and power factor. The proposed system eliminates the use of any mechanical or centrifugal switches which is located inside the motor. This avoids the possibility of the switch failure and leads to less operational and maintenance cost and improves the system reliability.

4. Performance evaluation In this section, the performance characteristics of the system under consideration will be evaluated and thoroughly investigated for given loading conditions. 4.1. Test machine To verify the proposed analysis and modeling of threephase induction motor operating from single-phase supply with an electronically controlled capacitor, an experimental prototype is implemented. A 1.8 kW, 220 V, 50 Hz, 7.8 A three-phase wound rotor induction motor connected in star for single-phase operation as shown in Fig. 1 and has the

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Table 1 Induction motor parameters Parameter

Symbol

Value

Stator resistance Rotor resistance Stator leakage reactance Rotor leakage reactance Magnetizing reactance Moment of inertia Number of poles Capacitor

Rs Rr Xls Xlr Xm J P C

2.53 () 0.90 () 3.01 () 0.6946 () 78.5 () 0.016 (Kg m2 ) 4 330 (␮F)

parameters shown in Table 1 [14–15] is used in this paper. The motor is mechanically coupled to a dc generator, which by adjusting its field voltage provides a variable load torque to the motor. The shaft speed is measured using a tachometer mounted on the same shaft couples the motor and the dc generator. 4.2. Results and discussions Based on the machine parameters described in Table 1, an optimal capacitive reactance achieving minimum unbalance and zero negative sequence voltage are demonstrated in Fig. 3. This figure shows that a small capacitive reactance XC ; large capacitor size; during the starting period and large XC ; small capacitor; during the running period for best running conditions are required. It must be evident that XC must be continuously adjusted to keep minimum unbalance or zero negative sequence voltage at different speeds. From Fig. 3, it can be seen that the capacitor required for minimum unbalance reaches its maximum value at stand still. After that point the capacitor related to minimum unbalance begins to decrease as the speed increases. It is clear that one can choose an optimal path to control the capacitor and achieve a minimum unbalance as shown in Fig. 3. However, it is difficult to realize a continuously adjustable mechanical capacitor. The proposed switched capacitor can change

Fig. 3. Optimal starting curve of the capacitive reactance as a function of motor speed.

Fig. 4. Effective capacitive reactance of a 330 ␮F controlled capacitor.

the effective capacitor and provides an effective method of overcoming the difficulty associated with the need for a continuously variable capacitor. Fig. 4 shows the computed and measured variations of the effective capacitive reactance for a 330 ␮F capacitor; switched in accordance with the principle proposed in this paper; with the duty cycle of the electronic switch. For practical use of the proposed controlled capacitor with the threephase induction motor operating from single-phase supply, a small fixed capacitor of 9.4 ␮F is connected in parallel with the controlled capacitor as shown in Fig. 1 to produce a path for the motor current of phase c during the off periods of the controlled switch and to complete the required starting optimal capacitor to 340 ␮F as depicted in Fig. 3. The experimental duty cycle required for the electronic switch to keep minimum unbalance in motor phase voltages at all operating speeds is shown in Fig. 5, which shows that the duty cycle is higher when the motor is near the standstill mode and then gradually decreases as the motor starts. When the speed of the motor closes to the synchronous speed, the duty cycle will reduce near zero value due to the fact that the

Fig. 5. Duty cycle as a function of the motor speed.

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Fig. 6. Simulated and experimental speed–time response at no load. (a) One capacitor, (b) two capacitors, (c) proposed controlled capacitor.

value of the optimal capacitor is decreased as the speed is increased as shown in Fig. 3. For the purpose of comparison, three different techniques for testing the same motor are studied. The motor was started and running with one capacitor value (330 ␮F), two capacitor values; one for starting (330 ␮F) and the other for running (50 ␮F); and with a 330 ␮F capacitor switched in accordance with the principle proposed in this paper. The effect of the capacitor on the motor performance will be studied in the following. Simulated and measured starting and steady state responses of the motor speed for the three cases at no load are shown in Fig. 6. While Fig. 7 shows the speed responses for the same three cases when the motor operating under a load of 3.0 N m. It can be seen that when the capacitor is switched in accordance with the principle proposed in this paper, the acceleration time becomes shorter compared to that of the other two cases and the decrease in starting time is more apparent at load condition as shown in Fig. 7. Apart from the good agreement between the simulated and experimental speed responses, it is noticeable that there are evident undulations in the motor speed in Figs. 6(a) and 7(a) because the running fixed capacitor is far from the optimal value. These speed ripples become less remarkable with two fixed and proposed switched capacitor. Steady state results of Figs. 6 and 7(a) and (b) give near results because the value of the running fixed capacitor is good selected to be near the optimal value.

Fig. 8 shows a comparison of computed and experimental supply current at starting for the same three cases under a load of 3.0 N m. Although the current is almost the same during the starting period, the current at steady state is still high for the one capacitor value, decreased when using the two capacitor values. However, when the capacitor is switched in accordance with the principle proposed in this paper, the steady state current is more reduced. Also, the starting period becomes shorter especially at load condition as demonstrated in Fig. 8(c). Fig. 9 shows the computed electromagnetic torque responses at starting for the three cases under a load of 3.0 N m. It can be noted that although the capacitor is not selected according to the criteria of minimum pulsating torque, the pulsating torque is reduced with the two value operating capacitor and more improved with the continuously controlled capacitor. Comparing between (b) and (c) of Figs. 6–9, one may notice that the advantage improvement in the motor performance using the proposed switched capacitor is achieved at starting only when compared the results with that of two fixed capacitor technique, this is because the value of the two fixed capacitors are carefully selected, specially the running one, to be near the optimal value required to give the criteria of minimum unbalance at the specified particular operating speed. If the speed is changed for any reason (such as speed control or even due to the loading conditions), the value of the fixed running capacitor becomes not suitable and another value of the running capacitor must be

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Fig. 7. Simulated and experimental speed–time response at a load of 3 Nm. (a) One capacitor, (b) two capacitors, (c) proposed controlled capacitor.

selected. The effect of the switched capacitor in the motor performance can be easily noticed by comparing between (a) and (c) at the steady state of Figs. 6–9 because the running fixed capacitor becomes far from the optimal desired value.

In view of the obtained results, it can be concluded that with continuously adjusting the capacitor size, it is possible to achieve very satisfactory transient and steady-state performance of three-phase induction motor operating from singlephase supply.

Fig. 8. Simulated and experimental stator current a load of 3 N m. (a) One capacitor, (b) two capacitors, (c) proposed controlled capacitor.

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performance at different speeds. Experimental results validated the analytical results and a good agreement between the simulated and experimental results is achieved. The results in this paper present a step toward the design of some optimal switched capacitor techniques.

References

Fig. 9. Simulated electromagnetic torque at a load of 3 N m. (a) One capacitor, (b) two capacitors, (c) proposed controlled capacitor.

5. Conclusions The design and implementation of a controlled capacitor for a three-phase induction motor operating from singlephase supply has been presented by using a fixed capacitor in series with an electronic switch. The proposed system eliminates the use of mechanical or centrifugal switches which is located inside the motor. This avoids the possibility of the switch failure and leads to less operational and maintenance cost and improves the system reliability. The optimum effective capacitor value can be on-line adjusted at any operating speed by periodically changing the duty cycle of the controlled switch to achieve minimum unbalance in phase voltages or any other optimization criteria to improve the motor

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