Towards Economic Single-Phase Motor

Studying of operation balance in single-phase induction motors is an issue of interest due to the need for reducing the power consumption and increasing the motors’ life. The paper focuses on improving the motor performance by balancing the stator phase operation for the most common-used connection diagrams of single-phase capacitor-run induction motors (SPCRIMs) and three-phase induction motors (TPIMs) operating from single-phase supply (SPS). Therefore, a mathematical model is used to balance the motor operation by varying the frequency supply voltage. Characteristics of balancing parameters are investigated, various methods of motor balancing are presented and comparisons were done among these balancing methods.


Introduction
Economic single phase motors are required nowadays since a huge amount of power is consumed due to the wide using of these motors in the life fields such as: domestic, agricultural, industrial fields and so on [1][2][3].
By improving the performance of single-phase motors, farms, petroleum wells, homes and faraway workshops having only a single-phase line do not need to install expensive three-phase lines or resort to expensive inverters or diesel pumps.Also, in many applications, it may be necessary to use a three-phase induction motor on a single-phase supply system.For example, technical and economic advantages have been found to initially install a single wire earth Return (SWER) system for rural electrification in remote and hilly regions [4].
For single-phase capacitor-run induction motor (SPCRIM) and three-phase induction motor (TPIM) operating from single-phase supply (SPS), full-load current may have almost unity power factor which reduces the power company transformers and distribution losses.At balanced motor operation, the efficiency of single-phase motors may exceed 90 percent.Thus, single-phase motor performance can be improved and become competitive to that of three-phase motor on a three-phase line.Using the SPCRIMs is the best choice to compete the three-phase motors; whereas the run capacitor can improve the motor efficiency, the starting torque and the power factor.Also, using of additional reactive elements leads to robust motor balancing to ensure the excellent performance of the motor [5].
As a matter of fact, SPCRIM and TPIM fed from SPS suffer from heating due to the elliptical field caused by asymmetry of phase loads [6].The non-uniform operation of stator phases of these motors is negatively reflected on the winding temperature, the power factor and the efficiency of the motor [7,8].Therefore, the elimination of the asymmetric action is of a great theoretical and practical significance.
The conventional connection diagrams of SPCRIMs and TPIMs operating from SPS [9,10] using a constant capacitance value in a stator circuit supplied by constant frequency voltage, are not capable to provide balanced operation of stator phase in the whole range of motor slip [11].This is due to the ellipticity of a rotating field, which takes a circular form only under certain conditions.In this case, balancing is possible only at a certain value of slip and load fluctuations will cause the unbalancing of the motor and produce heating in the motor windings [12].Eliminating the asymmetry of phase loads is possible by using the following methods: 1) Using one value phase-shifting capacitor and regulating the frequency of the power supply.
2) Inserting of external reactive impedance into the circuit of the SPCRIM or TPIM fed from SPS.This is the most suitable method to provide the required values of phase currents and appropriate angles between them (rigorous symmetry).
3) Switching a number of stator winding turns and regulating the value of phase-shifting capacitance [13,14], which is regarded as the most economical method from the point of view of electrical energy utilization and motor heating.This paper develops a mathematical model to balance the motor operation by varying the frequency of supply voltage and investigating the characteristics of the balancing parameters.Also, the paper presents advanced review for the used methods of balancing, comparison among them through behavior investigation and limitations of each method for the most used connection diagrams of induction motors fed from single-phase supply in practical applications.

Balancing of the Motor Operation by Controlling the Supply Frequency
The produced field in the SPCRIM and TPIM operating from SPS can have a rectilinear, an elliptical or a circular form, depending on the reactance of the phase-shifting capacitor.Of course, the machine will have the best efficiency and power factor when the field has a circular form.Thus, phase currents are equal in magnitude and the phase angle between them is 90 electrical degrees at SPCRIMs or 120 electrical degrees at TPIM operating from SPS.The reactance can be controlled by varying the frequency of the supply voltage and under certain conditions at which negative sequence current   2 I  becomes zero the motor operation is balanced [15].The capacitor reactance value that satisfies the first balancing condition can be calculated from the following relation: the second balancing condition is: where A and B are the balancing coefficients

Balancing Conditions of SPCRMs with Two Windings Connected in Parallel
Circuit diagram of SPCRIM with two windings connected in parallel is shown in Figure 1.Utilizing the symmetrical components methods, the unbalanced motor variables can be decomposed into positive (forward) sequence and negative (backward) sequence components [16,17].
Figure 2 shows the equivalent circuit of these components [18].According to the symmetrical components method, the phase currents can be written as [16,19] From Kirchhoff's Law, the voltages that model the SPCRM are where From Equations ( 5) and ( 6), the balance equation (at Substituting real and imaginary parts of the imped-Open Access EPE ances then gives Solving this equation yields From Equations ( 9) and ( 10), we get Thus, the balance coefficients are

Balancing Conditions of Three-Phase Induction Motor Fed From Single-Phase Supply
The circuit diagram of TPIM connected in delta and operating from SPS is shown in Figure 3. From Kirchhoff's Laws, the voltages and currents are Substituting the symmetrical components for voltages and currents into Equations ( 11) and ( 12) gives [20]: with a balanced condition By solving Equations ( 18) and ( 19), we get As a result, the balancing coefficients are Using the same procedures of analysis, balancing coefficientsfor the least circuit diagrams of SPCRIM and TPIM fed from SPS can be derived.Balancing coefficients are found to be as in Table 1.
Further, when the frequency is kept constant, Equation ( 2) is satisfied at certain value of slip.Varying of the slip (S) leads to variation of the stator currents while for certain values of the slip specifically S = S sym , the stator currents will equal each other [21].The phase-angle between the phase currents that required establishing the balance can be obtained by using shifting capacitor.In other words, for any slip (S) there is a certain frequency (f sym ) at which motor will be balanced.In order to find In parallel the freque at which a balanced op n of the motor eved for different values of slip, the value f R 1 and X 1 ust be found from the equivalent circuit of single phase motor in Figure 2(a) as Substituting R 1 and X 1 from Equation (20) and Equation (21) into Equation ( 2) and rearranging the obtained equation with neglecting of stator active resistance, the per-unit balancing frequency  can be found as: For low and medium power motors, one may consider r S X X   , and then the per-unit balancing frequency calculated as: when the frequency value is given, the slip at which the motor operation is balanced can be derived as: The critical slip (slip at maximum torque) is a function f frequency and can be calculated from the expression

Balancing the SPCRIM by Inserting an Inductive Reactance into the Stator Circuit
The values of the balancing impedance capacitive) for the most common connection diagrams of SPCRIMs and TPIMs fed from SPS can be determined (inductive and by the following group of equations [5]: where, the coefficients of Equations ( 26) and ( 27) can be obtained from Table 2.

Balancing the Motor Operation by Controlling the Capacitance Value
In this method, the frequency is constant and f y the balance operation when the load is changing.The balancing capacitor value can be g capacitance value can be calculated from the Equations ( 5) and ( 6) by equating requently is equal to the nominal frequency whereas the capacitance is varying to satisf controlled electronically [4,23].

Balancing of SPCRM with Two Windings Connected in Parallel
For the SPCRIM with two windings connected in parallel where the capacitor value is varying in respect with the load as in where

Simulation and Results
Curves of balancing parameters X K , S sym and S cr versus frequency ,depending on the Equations ( 1) , ( 24) and ( 25), were investigated by using labVIEW software for the SPCRIM and TIM operating from SPS with the following data: It can be seen from these figures that S sym is inversely proportional to the frequency, where its value at low frequencies approaches 1.This means that the motor can be started with a balanced status and this is considered a very important aspect in the intermittent periodically duty motors.However, at steady-state operation, the low frequency could cause a great loss in energy because of the high value of balancing slip, and this should be av   10.5 , 196.5 .
oided.The doted curves give the variations of critical slip versus frequency.It should be noted that as long as S > S sym , the motor will be stable and the stability will depend on the difference between S cr and S sym where the greater the difference the more stable the motor.Therefore, the steady-state region is defined when f > 0.2 f n .
The impedance characteristics of the balancing elements are also built by using a labVIEW software.
Figure 9 shows the relation between the balancing impedance and the slip at different frequencies for the above described motors with the attached connection diagrams: The values of reactance are calculated by using Equations ( 26) and ( 27) for Figures 9(a) and (b), respectively.Figure 9(a) shows that the inductive reactance X L is high at no-load condition, and decreases by increasing the load till it reaches a minimum value without crossing the X-axis (only inductive behavior.This is clear for high  frequencies of the supply voltages.The balancing capacitive reactance X K is high at no-load condition and decreases with increasing the load in the same way for all the frequencies of the supply voltages.

Table 2. Coefficients of balancing equations for common types of circuit diagrams.
Figure 9(b) shows that both the balancing reactance X L and reactance X K have the same behavior.At first, they increase by increasing the load until they reach maximum values, then they begin to decrease again.Balancing inductive reactance X will cross the -axis g the supply voltage frequency, the crossing point of the X L with the X-axis will occur at lower values of the slip.It is clear that at high frequencies, the balanced operation will be achieved  by regulating the capacitance value only, in other words, both balancing elements should be capacitors.
The same inductive and capacitive behaviors are occurred for the least of connection diagrams, listed in the Table 2, based on the group of Equations ( 26) and ( 27The balancing capacitan haracteristic was built using Equation (28) also for the motor of power P n = 2.8 kw as shown in Figure 10.
This figure shows that for constant frequency f = f n , the balancing capacitance value is proportional to the slip until specified value then the relation becomes nonlinear and the balancing capacitance almost hasn't considerable change as slip increases over the critical slip.The bal-).ce c The advantages of the balancing method by changing the frequency are: 1) Wide range of speed control.
2) Soft speed control and improvement of starting characteristics.
3) This method can be used for various power rating motors with any stator circuit connection.
The disadvantages of this method are: 4) Steady state motor operation at low freque ies than for the run condition.Although increasing the capacitance over the nominal value helps in balancing but it s accompanied by increasing the curre i the auxiliary winding .Therefore, this method is promising for the variations of load around the nominal value if the motor duty is continuous.

Conclusions
e study discusses the various methods to improve the performance of SPCRIMs and TPIMs operating from SPS. Single-phase induc neering practice and spends a lot of electricity each year.The promotion of induction motor's efficiency has great significance for energy consumption, so the optimization design of single-phase induction motor is necessary.The mathematical model seems to be well.
The mathematical model is used to balance the motor nc (F < 0.2) is forbidden due to the small maximum to the nominal torque ratio.
5) Expressions of balancing frequency, slip and capacitive reactance are awkward and have high orders.
Balancing by varying capacitance value of the condenser at constant frequency is the most economic especially if it is realized by electronic way but this method is not fair as the slip goes far from the nominal value.
For having robust balancing, in addition to the shifting phase capacitor, reactive elem the stator circuit.This method ent should be inserted into will reduce the heat generring the steady state operation mode speed control.Thus, the benefits of egardle masin from Al-B A bers of Switches," IEEE Transactions ated in the motor du for the full range of this method include improved power factor, energy saving and elimination of the need for additional winding taps for speed variation.
According to the generalized calculation equations for balancing element impedance, the connection balancing diagrams could be grouped into two groups.For the first group of connection diagrams, the behavior of the balancing element is inductive over the whole slip r ss of the value of the voltage frequency.While for the second group of connection diagrams, the behavior of the balancing impedance X L will become capacitive depending on the load and the voltage frequency.

Figure 1 .Figure 2 .
Figure 1. Circuit diagram of SPCRM with two windings connected in parallel.

Figure 3 .
Figure 3. Circuit diagram of three-phase Δ -connected moor fed from single-phase supply.t

Figure 4 ,
the balancin the absolute values of A I  and B I  as [10,24].

Figures 5 - 8
Figures 5-8 show the obtained curves for the most used connection diagrams.It can be seen from these figures that S sym is inversely proportional to the frequency, where its value at low frequencies approaches 1.This means that the motor can be started with a balanced status and this is considered a very important aspect in the intermittent periodically duty motors.However, at steady-state operation, the low frequency could cause a great loss in energy because of the high value of balancing slip, and this should be av n

Figure 4 .Figure 5 .
Figure 4. Single-phase induction motor with two windings connected in parallel and electronically controlled capacior.t

Figure 8 .
Figure 8. Balance of Ү-connected TPIM three-phase induction motor fed from single-phase supply.

Figure 6 .Figure 7 .
Figure 6.Balance of SPCRIM with two windings connected in series.

Figure 9 .
Figure 9. Impedance characteristics; (a) auxiliary winding is shunting by inductive element, (b) Inductive element is inserted in series with the main winding.

Figure 10 .
Figure 10.Balancing capacitance of single-phase capacitorrum induction motor versus slip.ancingcapacitance for the start condition is much larger