Improvement of Torque Production in Single-Phase Induction Motors

*Corresponding author’s e-mail address: baolarinoye@abu.edu.ng doi: http://dx.doi.org/10.4314/njtd.v14i2.1 ABSTRACT: Existing single phase induction motors exhibit low starting torque. Moreover, during accelerating time and at steady state, they produce a significant level of torque pulsations which gives rise to noise and vibration in the machine. As part of efforts to mitigate these problems, a performance improvement strategy using a PWM inverter to drive the existing motor is implemented in MATLAB/Simulink environment in this work. The drive supplies variable voltage and phase to the auxiliary winding with the aid of a pulse width modulation (PWM) technique and a PID controller. Simulation results show the starting torque of the motor increased by 75% under the developed drive scheme. In addition, torque pulsations reduced from 1.4 Nm peak-peak to 0.14 Nm peak-peak at steady state. It was observed that the accelerating time reduced by 30% compared to the accelerating time under line operation. The strategy eliminates the need for series-connected capacitors thereby potentially enhancing the reliability of the motor.


I. INTRODUCTION
Single phase induction motors are motors fed from a single phase power source.Power is supplied to the rotor of these motors by electromagnetic induction.The different types of single phase induction motors (SPIMs) include the split phase, shaded pole and the capacitor types.The most common of these is the capacitor-run single phase induction motor (CRSPIM).This type of SPIM has a capacitor connected in series with the auxiliary winding and its purpose is to balance the currents in the main and auxiliary windings during running conditions thereby providing quiet operation and maximum efficiency.Often times, another capacitor of higher capacitance is provided in the auxiliary winding circuit to ensure high starting torque.
This starting capacitor, however, is disconnected by a centrifugal switch after the motor attains about 75% of its rated speed.Capacitor-run single phase induction motors find application in residential, commercial and industrial centers across the globe.They are used in washing machines, dish washers, fans, refrigerators and air-conditioners just to mention a few.The problem with these motors are (i) high torque pulsations which give rise to noise and losses and (ii) low efficiency especially when used under non-rated conditions.A lot of work aimed at addressing these problems has been reported in literature.
The aspiration in most of the methods used to address these problems is the elimination of the double unbalance in supply voltage/current and in the main and auxiliary windings.Chomat and Lipo (2003) applied a half bridge inverter to supply an auxiliary winding voltage shifted by 86 0 from the phase of the main winding voltage to achieve balanced flow of winding currents.Asghari and Fallah (2012) proposed an inverterbased method to produce the necessary phase shift required for motor operation.CRSPIM performance under different capacitance values was investigated in Hekmati et al. (2014).
The study focuses on a control strategy in which a parallel bidirectional switch plus a fixed capacitor was simulated to reduce torque pulsations in the motor.In this paper, effort is made to supply balanced voltages to the windings of an existing CRSPIM with a view to reducing pulsations in the electromagnetic torque produced in the motor at all operating points.

II. CONDITION FOR ELIMINATING TORQUE PULSATION
The average or useful torque in the single phase induction machine going by the double revolving field theory (DRFT) (Morrill, 1929;Collins et al, 1988;Kim et al, 2003) is given as; The pulsating component of the torque produced is given as (Morrill, 1929;Collins et al., 1988): current magnitudes respectively, is the difference between the main and auxiliary winding current phase angles, and are the equivalent forward field resistance and reactance respectively while and are the equivalent backward field resistance and reactance respectively.These parameters are defined as follows (Morrill 1929;Collins et al. 1988;Jang 2013): (3) (4) (5) where , and are the magnetizing reactance, rotor resistance and reactance referred to the stator.s is the slip.In literature, the magnitude of the pulsating torque component is typically used to characterize the strength of the torque pulsations (Vaez-Zadeh and Langari, 2000).This magnitude is obtained from eqn (2) and is expressed as (Collins et al, 1988;Morrill, 1929): Equation ( 7) suggests that the pulsating torque can be eliminated if the following expression is true; (8) Equation ( 8) constrains the magnitude of the main and auxiliary windings to the following relationship for equal ; (9) Equation ( 9) will be used in section V to compute the magnitude of the voltage which when applied to the auxiliary winding voltage could eliminate torque pulsations in the single phase motor.

III. MODEL OF THE CRSPIM
The model equations of the capacitor-run single phase induction machine expressed in the stationary reference frame (Ong, 1998) are given as follows; (10 Derivative operator given as The qaxis stator winding represents the main winding while the daxis stator winding represents the auxiliary winding.The d-q axis is a fictitious axis used to simplify the derivation of the motor equations.

IV. PWM INVERTER DRIVE FOR PERFORMANCE IMPROVEMENTS
The inverter drive comprises an inverter topology and associated controller that will enable the supply of a voltage to the auxiliary winding whose phase lead is as close to 90 0 as possible with respect to the main winding voltage.The single phase inverter drive is shown in Figure 1.In this figure, the main ac voltage is rectified to dc and then the rectified dc voltage is filtered before feeding the inverter.The speed signal was used to calculate a reference voltage.This reference voltage is compared with the actual voltage supplied to the auxiliary winding and the error signal is processed by a PID controller to generate the PWM signals required to switch the devices of the inverter in order to synthesize the desired output to be injected into the auxiliary phase winding.This output is a voltage whose amplitude and phase is proportional to the machine speed in compliance with the condition for eliminating torque pulsations in the motor.The scheme was realized in MATLAB/Simulink environment.It was applied to a motor whose data is provided in Table 2.
For the purpose of analysis, the DC side of the inverter is supplied with 200 V and the modulating index is chosen in such a way that the synthesized output voltage is equal to the rated voltage of the existing SPIM.The cut-off frequency, of the output filter relates to the filter values (Pawar and Kulkarni, 2015) by the following equation; is chosen as 600 Hz which is a compromise between eliminating higher order harmonics of the output voltage and keeping the sizes of the output inductor, and capacitor, small.
is chosen as 0.04 mH and is then calculated using eqn (22).The component values are provided in Table 1.The modulating signal in this scheme is the output of the PID controller.The inverter switches are pulse width modulated with this modulating signal and a carrier triangular wave having a frequency of 2 kHz.The switching frequency of 2 kHz was chosen because it helps, along with the filter components, to remove the harmonics present in the output voltage of the inverter phases while minimizing the switching losses in the inverter.

V. COMPUTATION OF REFERENCE VOLTAGE
Using the DRFT with the SPIM equivalent circuit as given in Figure 2 (Collins et al., 1988), the voltage equations of the main and auxiliary windings may be obtained as follows; ))+ ( 23)   The drive scheme in Figure 1 was modelled and the PID controller parameters were determined by a trial and error process using the MATLAB pidtool command (Control system toolbox user's guide, 2013a).The values obtained for the proportional, integral and the derivative gains are 0.07646, 4.6678 and 0.00031312 respectively.

VI. SIMULATION RESULTS AND DISCUSSION
Simulation results of the electromagnetic torque, reference voltage, main and auxiliary winding voltages and the speed responses are presented in this section.Figure 3 gives the dynamic torque production of the motor when the topology of Figure 1 was applied.The total simulation run time is 3s and a rated load of 1Nm was applied at a time of 2s.
It can be observed in Figure 3 that the average starting torque is approximately 7 Nm and that the torque pulsations have reduced significantly when compared to those of the line-operated capacitor-run machine given in Figure 4.In Figure 4, the starting torque of the line-operated capacitor-run motor is seen to be 4 Nm.The point to note is that the inverter-driven motor has a value of starting torque that is 75% higher than that of the line operated capacitor-run motor.In addition, the peak to peak magnitude of the torque pulsations in the existing CRSPIM is 1.4 Nm at rated load.This value was reduced to 0.14 Nm in the inverter-driven motor as observed in Figure 3.This reduction represents an improvement of 71% over the level of torque pulsations in the existing CRSPIM.Figure 4 was obtained by solving for stator and rotor currents in eqns ( 10) -( 19) and then using the currents to calculate the electromagnetic torque in eqn (20).(Olarinoye and Oricha, 2013).
The actual inverter output voltage is superimposed on the reference voltage in Figure 5 for ease of comparison.It can be observed that the output voltage is identical to the reference voltage.This is due to the PID controller action.The voltage is seen to vary in amplitude as motor accelerates from rest to steady state, between 0 and 3 s.The actual inverter output voltage is seen to track the reference faithfully from a time of about 0.02 s.The import of this reference tracking is seen in Figure 7. Figure 7 shows the main winding voltage and the auxiliary winding voltage which is the same as the inverter output voltage plotted together for the period between 1.1 s and 1.14 s.It can be seen that the auxiliary winding voltage leads the main winding voltage by a time phase angle of 90 0 and that its amplitude is about 1.18 (i.e.winding turns ratio) times greater than that of the main winding voltage.This result verifies the analysis given in section V.The controller is therefore able to command the inverter to supply balanced voltages to the stator windings of the single phase induction motor.This controller action in conjunction with the action of the pulse width modulated inverter explains the reason why there is a significant reduction in the torque pulsations of the machine as seen in Figures 3 and 4. The graph of Figure 8 shows the speed response of the inverter-driven motor.It clearly shows that the motor accelerates from rest and reaches steady state speed in 0.7 s.The speed is seen to drop to 1730 rpm at a time of 2 s after the load was applied.The gain in average starting torque increases the acceleration time of the motor with the consequence that it reaches steady state at time 0.7 s in Figure 8, down from 1s in Figure 9 for the line-operated capacitorrun motor.This reduction represents a 30% improvement in the accelerating time of the motor.The broader implication of the reduction in acceleration time is that the machine runs up to steady state more efficiently.

VII. CONCLUSION
The torque performance of an existing SPIM under line operation has been compared with those of the same motor under inverter operation.In this paper, the inverter supplied voltage to the auxiliary winding with the aid of a PWM control scheme.Simulation results show a 75% increase in starting torque for a 110 V, 60 Hz CRSPIM.Results also show a 71% reduction in the magnitude of torque pulsations in the motor.These results are based on the comparison between the performances of the motor when driven by the inverter and the performances of the motor when driven conventionally by a fixed voltage source and with seriesconnected capacitors in its auxiliary winding circuit (i.e.line operation).The results show that the inverter-driven motor is potentially able to operate more quietly, reliably and efficiently.It is recommended to determine the viability of the inverter drive scheme by practically implementing the scheme and conducting a cost/benefit assessment in order to determine whether or not the benefits outweigh the additional cost of inverter and its associated control scheme.
in the main winding ds i -Current in the auxiliary winding referred to main winding qs  -Flux linkage of the main winding ds  -Flux linkage of the auxiliary winding referred to main winding.-Voltage applied across the qaxis rotor winding referred to main winding -Voltage applied across the daxis rotor winding referred to main winding qr i -Current in the q -axis rotor winding referred to the main winding dr i -Current in the d -axis rotor winding referred to main winding qr  -Flux linkage of the q -axis rotor winding referred to the main winding.dr  -Flux linkage of the d -axis rotor winding referred to main winding qs r -Resistance of the running winding ds r -Resistance of the starting winding referred to main winding.inductance between the q -axis stator and rotor windings md L -Mutual inductance between the d -axis stator and rotor windings referred to main winding.lqs L -Leakage inductance of the main winding lds L -Leakage inductance of the auxiliary winding referred to main winding lr L -Leakage inductance of the rotor winding referred to the main winding.

Figure 2 :
Figure 2: Equivalent Circuits of the Capacitor-Run Single-Phase Induction Machine.

Figure 4 :
Figure 4: Dynamic Torque Production in the CRSPIM.

FigureFigure 5
Figure 3: Dynamic Torque Production in the Inverter-driven Motor.

Figure 6 :
Figure 6: Reference and Actual Auxiliary Winding Voltage between 0 and 0.1s.

Figure 7 :
Figure 7: Main Winding Voltage and Auxiliary Winding Voltage in 2 Periods.

Figure 8 :
Figure 8: Speed vs Time Characteristics of the Inverter-Driven Motor.

Figure 9 :
Figure 9: Speed vs Time Characteristics of the line-operated Capacitor-Run Motor.