Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 5131696 | https://doi.org/10.1155/2019/5131696

Vasilija Sarac, Tatjana Atanasova-Pacemska, "Multiparameter Analysis for Efficiency Improvement of Single-Phase Capacitor Motor", Mathematical Problems in Engineering, vol. 2019, Article ID 5131696, 13 pages, 2019. https://doi.org/10.1155/2019/5131696

Multiparameter Analysis for Efficiency Improvement of Single-Phase Capacitor Motor

Academic Editor: Yongheng Yang
Received02 Jan 2019
Revised03 Apr 2019
Accepted16 Apr 2019
Published22 May 2019

Abstract

Single-phase motors are known for their small power ratings and their usage in various household appliances. Although they are not large electricity consumers, their widespread application contributes to the overall electricity consumption. In addition, standard IEC 60034-30-1:214 defines the efficiency levels for single- and three-phase motors and stipulates the increased electrical efficiency for the electrical motors. Therefore, this paper sets the parametric analysis of permanently split capacitor motor with five different design parameters that have impact on the efficiency of the motor. As an output from the parametric analysis, two different optimized motor models are obtained with increased efficiency. The impact of each parameter on motor efficiency—as well as on the other operating characteristics, like starting torque, overloading capacity, rated current, starting current, total losses, and power factor—is analyzed and adequate conclusions are derived. The obtained motor models are verified with Finite Element Method (FEM) for magnetic flux density distribution.

1. Introduction

Capacitor motors belong to the single-phase motors that have a capacitor in the auxiliary winding usually aimed for motor starting. In the case of the permanently split capacitor motor, there is only one capacitor, placed in the auxiliary winding, which is permanently in operation, during starting as well as during motor running. In general, the single-phase motors have low power ratings but they are widespread in the household appliances and everywhere where single-phase supply is available. This makes the electricity consumption from the single-phase motors a considerable portion of the overall electricity consumption in the households and industry. In March 2014, the standard IEC 60034-30-1 has been published. It defines four efficiency classes for induction motors (single- and three-phase), IE1 standard efficiency, IE2 high efficiency, IE3 premium efficiency, and IE4 super premium efficiency (Figure 1). From January 2017 the legally specified minimum efficiency IE3 must be maintained for power ratings of electrical motors from 0.75 kW up to 375 kW. Improving the motor efficiency has become an imperative for electrical motor producers. Nevertheless, the production costs should be maintained on competitive level through analyzing the various optimization aspects of the motors.

The optimization of the electrical motors with respect to the efficiency improvement based on the evaluation of the mechanical parameters and scaling laws at three-phase induction motors has been analyzed [2, 3]. Optimization of single-phase capacitor motors has been investigated with triac-based voltage controller or in the drive configuration with Hall sensor [4, 5]. Over the years, different optimization algorithms have been employed for the optimization of the single-phase motors such as particle swarm optimization, surrogate filed-circuit model, or genetic algorithms applied in single- or multiobjective optimization models [69]. Hybrid algorithm consisting of the particle swarm optimization and the genetic algorithms was applied in determining the optimal operating point of the two-phase motors regarding the increased efficiency and reduced torque ripple [10]. The last one presents the optimal motor operating point after optimization has been done and efficiency factor has been calculated by FEM. Artificial neural network controller has enhanced the efficiency and the dynamic responses of three-phase induction motors particularly when the motors are operated at partial load [11]. Optimization of the induction motors advances further with the usage of fuzzy logic based controller acting as a supervisor in reducing flux levels during the transients and contributes to the efficiency improving of the dynamic operating modes [12]. Maximization of the efficiency at the induction motors is achieved by introducing mechanical improvement in stator and rotor core, i.e., slits in the middle of the stator and the rotor teeth [13, 14]. This paper sets the comprehensive parametric analysis regarding efficiency improvement of permanently split capacitor motor and gives an overview of how the parameters are changed in optimized models in comparison to the starting model. The variations of all important operating parameters like starting and maximum torque, rated and starting current, current in main and auxiliary winding, losses in all motor parts as well as total losses, input and output power, rated torque, and power factor are also presented. Particularly the influence of each design parameter, which varies within certain limits, on the efficiency of the motor is presented. Consequently the paper gives major guidelines for motor designers about how the different design parameters (motor length, number of conductors of the main and the auxiliary winding, rotor diameter, and rotor slot opening) have influence on complete motor behavior during various operating modes: rated load, no-load, motor starting, and break-down operation. In addition, motor operation is evaluated by taking into consideration how maximization of the efficiency affects other operating conditions. Advantages and limitations of the proposed optimized models are also discussed. The starting motor model (BM) is a single-phase permanently split capacitor motor with rated data: voltage 220-240 V, frequency 50 Hz, rated current 1.32 A, rated power 124 kW, product of the company MicronTech. Calculation of the operating characteristics of the single-phase capacitor motors is based on the method of symmetrical components and the double field revolving theory [15, 16]. Often these analytical calculations require a lot of time due to the complex mathematical model of the motor that is becoming even more complex when the optimization is in question. In this paper Maxwell software for automated computing of motor performance is used allowing fast calculation of motor parameters and characteristics for various operating models and modes. The accuracy of the obtained results is highly dependent on the accuracy of the software model of the motor. Therefore, the starting motor model (BM) is verified by comparing its output results with available data from the motor’s producer. Once the computer model is verified as sufficiently accurate, the parametric analysis is set by varying four different motor parameters in the first optimized motor model (M1): number of conductors in the main and the auxiliary stator winding, the motor axial length, and the outer diameter of the rotor. One more parameter is added in the second optimized motor model (M2)—the length of the rotor slot opening. The influence of the rotor slot distance from the rotor surface on the motor operating characteristics has been analyzed leading to the conclusion that motor performance is improved as the slot distance decreases [17]. The effect of the manufacturing process, which contributes to the larger fill factor of the rotor slots and its correlation with the motor efficiency, was investigated as well [18]. Closing the rotor slots raises additional questions about motor losses and core saturation in rotor slot bridges [19, 20]. The closed rotor slots have significant impact on the no-load losses mainly due to the reduction of the losses at the slot upper part [21]. Therefore, the rotor slot opening is optimized in the second model leading to the motor construction with closed rotor slots. Both optimized models have increased efficiency in comparison with the starting model. Nowadays, FEM analysis is inevitable part of the motor design procedure as it verifies the feasibility of the proposed motor design in terms of magnetic core saturation at critical points of the motor construction [2224]. The motor design procedure is completed by FEM analysis of the magnetic flux density distribution in the models cross-sections.

2. Mathematical Theory for Single-Phase Motor Modeling

Despite its simple construction, the mathematical model of the single-phase motors, which describes the various motor operating modes, is quite complex due to the existence of the rotating elliptical electromagnetic field in the motor air gap. This air gap electromagnetic field is highly unsymmetrical; the application of the well-known theory and the mathematical models for the three-phase symmetrical induction motors is improper and inaccurate. Therefore, the unsymmetrical magnetomotive force (mmf) currents and voltages corresponding to the two windings of single-phase induction machines may be decomposed in two symmetrical systems (Figure 2) which are the forward and backward components of the two-phase system [15].

Here following relations are valid:where and are general variables associated with the auxiliary winding and main winding, respectively.

The superposition principle yields the following.

The first step in the motor analysis is to determine all motor parameters: , main stator winding resistance; , main winding leakage reactance; , auxiliary stator winding resistance; , auxiliary stator winding leakage reactance; , magnetizing reactance; , rotor winding resistance; and , rotor winding leakage reactance [15]. Calculation of the motor parameters is based on the motor geometry obtained from the producer.

Based on the method of symmetrical components, impedance parameters , , and are calculated, necessary for obtaining the symmetrical components of the stator currents and electromagnetic torque [15]:where s is the motor slipwhere C is the capacitance in the auxiliary winding and a is the ratio of the winding turns of the main and auxiliary stator winding. The symmetrical components of the currents of the main stator winding are calculated fromwhere Vs is the motor power supply voltage.

The supply line current is found from the following.The currents in the main winding and the auxiliary stator winding are calculated, respectively.Components of the electromagnetic torque-direct and inverse are found:where p is the pair of the motor poles.

Electromagnetic torque is as follows.Motor input power is as follows.The power factor becomes as follows.The motor output power P2 is obtained from the input power P1 after subtracting all motor losses , which can be divided into several groups: stator and rotor ohmic losses, capacitor losses, iron core losses, and mechanical losses.

Stator ohmic losses are calculated at temperature of 75°C as the sum of the ohmic losses in the main and the auxiliary winding.Capacitor losses can be found fromwhere is the capacitor impedance .

According to [15] rotor ohmic losses at rated load can be approximately calculated aswhere andCore losses are mainly eddy current and hysteresis losses that in analytical form can be expressed as follows.

is the eddy current loss constant.

is the maximum flux density.

is the hysteresis loss constant.

is the material dependant Steinmetz constant (1.6 to 2).

Mechanical losses are mainly covering friction and windage losses. These losses can be determined by driving the motor at the rated speed with no-load or excitation. For the machines that operate at constant or nearly constant speed, these losses are constant. For an accurate calculation of motor losses, it is necessary to have an accurate calculation of motor parameters (resistances and reactances) as well as the currents in all motor windings. In addition, the magnetic flux density in the motor cross-section should be calculated, a parameter that is often more or less accurately predicted in motor analytical calculations. This emphasizes the need for an accurate computer-aided design of the motor capable of handling various design modifications, necessary for improving the motor efficiency factor (η), which can be finally obtained from the ratio of the output power P2 and the input power P1.

3. Computer-Aided Design of the Motor Models

Accurate modeling of the motor in a software program for automated computing of parameters and operating characteristics of the single-phase capacitor motor is an important step in the parametric analysis for efficiency improvement. Exact motor geometry is modeled with respect to the magnetic core dimensions and properties of the core material, i.e., magnetization curve B=f(H), with B being the magnetic flux density and H being the magnetic field strength (Figures 3 and 4).

Properties of core laminations are defined in accordance with lamination type DL 80, RK502.4. Electric properties of the materials are modeled as well. Stator winding, consisting of main and auxiliary winding made of copper, is placed in the main and the auxiliary slots. Rotor winding is of squirrel cage type, made of aluminum. Motor is aimed for constant power operation and it is modeled in the software program Maxwell using available data from the motor’s producer. The accuracy of the developed starting model (BM) is verified by comparing the output results from the model and available data from the producer. Table 1 presents this comparison.


BMProducer data

Rated power (W)124124
Rated torque (Nm)0.4160.411
Rated speed (rpm)28462880
Rated current (A)1.351.32
Maximum power (W)210210
Maximum torque (Nm)0.830.8

Once the starting model has been verified as sufficiently accurate, the parametric analysis is set by varying several motor parameters within the predefined boundaries. The variation ranges of the parameters are presented in Table 2.


Ranges of variationStep

Nr. of conductors in main slot (/)110÷1241
Nr. of conductors in auxiliary slot (/)125÷1351
Rotor diameter (mm)65÷65.60.1
Motor core length (mm)32÷371
Rotor slot opening (mm)0÷1.50.75

As a first step in the parametric analysis, four varied parameters were defined: the number of the conductors in the main and the auxiliary stator slot, length of magnetic core, and rotor diameter, i.e., the length of the air gap between the stator and the rotor. The parametric analysis resulted in 8085 different combinations that were solved. Out of these solutions, the best result or the best motor model in terms of the maximum efficiency at rated load was selected (model M1). The second parametric analysis was defined by adding one more varied parameter, i.e., the length of the rotor slot opening, and by taking into consideration the previous four varied parameters. This second parametric analysis resulted in 17325 different combinations. Their solutions gave the second motor model (M2) with better efficiency than BM and M1. In the next step, motor models are verified by FEM for magnetic flux density distribution. The flux density B is calculated from the magnetic vector potential A. In order for Maxwell’s equations to be solved, the complete machine cross-section is divided into numerous elements forming the mesh of finite elements (Figure 5). The electromagnetic field problems are solved by solving Maxwell’s equations in a finite region of space with appropriate boundary conditions and user-specific initial conditions in order to obtain solution with guaranteed uniqueness.where E is the electric field, D is the electric displacement equal to εE, ε is the permittivity, B is the magnetic flux density, and H is the magnetic field, equal to the product of material magnetic permeability and magnetic flux density, μB. J is the current density, σE, where σ is the conductivity. ρ is the charge density.

The FEM discretization of the domain of the analyzed object produces a set of matrix differential equations. They are solved with the time decomposition method (TDM). The domain is decomposed along the time axis and all time steps are solved simultaneously instead of solving them time step by time step. The nonlinear matrix equations are linearized for each of the nonlinear iterations. As an output from the FEM model, motor torque is calculated for all motor models. The comparison of the obtained torque from all FEM models with the torque from the parametric analysis verifies the accuracy of the FEM model.

4. Results and Discussions

4.1. Parametric Analysis

The output results from the parametric analysis give an overview of all important motor parameters and data. Motor parameters of all three models (BM, M1, and M2) are presented in Table 3 together with the output results from the parameters variations that resulted in the best motor models in terms of efficiency. All parametric analyses are run for constant power operation; i.e., all motors should have the same output power. The type of the winding is two-layer sinusoidal winding.


Motor parameterBMM1M2

Nr. of conductors per layer in main winding - (/)117124124
Nr. of conductors per layer in auxiliary winding - (/)130125125
Core length-L (mm)353737
Rotor diameter- (mm)65.465.665.6
Rotor slot opening –RSO (mm)1.51.50
Main winding resistance at 20°C -(Ω)25.37627.1627.16
Auxiliary winding resistance at 20°C-(Ω)69.1467.0567.056
Main winding reactance-(Ω)16.2419.1819.38
Auxiliary winding reactance-(Ω)24.6623.824
Rotor resistance-(Ω)22.3125.6625.98
Rotor reactance-(Ω)3.534.8121.525
Magnetizing reactance-(Ω)387.18702.3784.57
Capacitance-C (μF)666

For the best motor model in terms of efficiency besides motor parameters, motor operation in three typical operating modes, i.e., the rated load, the locked rotor, and the no-load, is analyzed. Table 4 provides the motor operating characteristics for the rated load. Here, an overview of all motor losses is presented as well. Mechanical losses (frictional and windage losses) were calculated as 9% of the total losses as often they are within range of 8-12 percent of the total losses. The no-load operating characteristics are presented in Table 5 together with the motor locked rotor (starting) characteristics. Detailed presentation of motor operation for different speeds is presented in Figures 6, 7, 8, and 9. Figure 6 presents the input (line) current for different speeds.


Motor characteristicBMM1M2

Rated speed (rpm)2846.482831.722831.72
Line current (A)1.351.17881.159
Main winding current density (A/mm2)4.4223.65623.608
Auxiliary winding current density (A/mm2)3.63.383.284
Main winding current (A)1.01660.840.829
Auxiliary winding current (A)0.36930.3460.336
Capacitor losses (W)72.37863.5459.98
Iron core losses (W)4.4873.933.13
Stator ohmic losses (Ω)43.34533.08331.9
Rotor ohmic losses (Ω)15.9116.60316.34
Mechanical losses (W)13.42413.2813.28
Total losses (W)149.545130.454124.67
Input power P1 (W)273.552254.352248.7
Output power P2(W)124123.89124.04
Efficiency factor (eta) η(%)45.33248.749.87
Output torque T (Nm)0.4160.41780.418
Maximum torque (Nm)0.830.7750.754
Power factor (/)0.9190.980.975
Rated slip (/)0.0510.0560.056


Motor characteristicBMM1M2

Main winding reactance (Ω)15.9618.718.7
Auxiliary winding reactance (Ω)24.3123.3723.4
Rotor reactance (Ω)3.244.347.59
Magnetizing reactance (Ω)283.34524.48552.5
No-load line current (A)1.150.67990.653
No-load current main winding (A)0.9530.4230.392
No -load current auxiliary winding (A)0.4130.390.386
No-load iron core losses (W)5.274.7873.82
No-load input power (W)156.244118.96114.97
No-load power factor (W)0.6140.7950.8
No-load speed (rpm)2985.912986.72986.7
Starting torque (Nm)0.1160.110.111
Starting line current (A)4.0773.6583.53
Starting current main winding (A)3.93253.5193.4
Starting current auxiliary winding (A)0.27040.27120.272

From Table 4 it can be observed that motor line current is decreased in both optimized models. The decrease of the line current is mainly the result of the increased resistance of the main winding due to the increase of the number of conductors in the main winding. The decreased line current leads to the decreased input power P1 or electricity consumption (15). The decreased input power affects the motor efficiency factor. The decrease of the current resulted in decreased ohmic losses in the stator windings (17). The parametric analysis is done with the motor output power as a constraint; i.e., output power should be kept constant which implies almost unchanged output torque as the relation between the output power and the torque is directly proportional. Some variations of the rated speed in different motor models that affect the rated output torque are possible, but in these particular models, they are not so emphasized. By taking into consideration that efficiency factor is defined with the ratio of the output and the input power, keeping the output power constant is forcing the decrease of the input power and the motor losses in order to achieve better efficiency factor than in the starting model. The smaller the motor losses, the better the obtained efficiency factor. As the electricity consumption of the motor is decreased, this leads to an energy efficient motor. This is the case with M1 and M2, with both models having smaller total losses than BM.

From Table 4 and the presented diagrams in Figure 7, it can be concluded that in the optimized models the power factor is increased. The increase of the power factor is a result of the decreased air gap due to the increased rotor diameter. In general, based on the designing experience, recommendation for the air gap length is [15]where g is the air gap length and is the rated output power. Therefore, by taking into consideration (27) and the motor rated power, it is estimated that the air gap can be further reduced. In general, the air gap between the stator and the rotor is a critical part of the motor construction. The length of the air gap affects the performance parameters like magnetizing current, overload capacity, cooling, and noise. A too small air gap would produce large space air gap harmonics and additional losses, while a too large one would reduce the power factor and the efficiency [15]. The larger air gap has some advantages and disadvantages. Among the advantages are increased overload capacity, increased cooling, reduced unbalanced magnetic pull, and reduced noise. The disadvantages are the increased magnetizing current and the reduced power factor. As expected, the optimized models are behaving in accordance with the aforementioned general rules. The overload capacity of optimized models is decreased (decreased maximum torque, Table 4) and the no-load current is decreased as well, due to the smaller air gap. The power factor and the efficiency are also increased. The influence of the current density of the stator windings on the overall motor design is analyzed as well. As a general recommendation, the current density of the stator windings should be within 3-5 A/mm2. The higher values of current densities have some advantages like reduction in the copper cross-section, the weight, and consequently the cost of the machine. The disadvantages are increase of the resistance, copper losses, and temperature rise followed by the reduction of the efficiency. As the models M1 and M2 have the decreased current density compared with the BM, they have lower resistance, decreased copper losses, and increased efficiency (Table 5). Figure 8 presents the output torque for all three motor models. As expected, due to the almost unchanged output power P2 and slight variations of rated speed , in all motor models, the output torque remains almost unchanged in all models for the rated load mode. The output torque and the output power are related to the following well-known equation.The efficiency factor for various speeds and for all motor models is presented Figure 9. The third model has the highest efficiency due to the last varied parameter, the length of the rotor slot opening. Generally, semiclosed slots or closed slots with very small or narrow openings are used in the construction of the rotor. The rotor with closed slots affects the motor performance in the following way: the rotor surface is smooth at the air gap, motor draws lower magnetizing current, and the noise and the starting current are reduced. In our models (M1 and M2), the no-load current is decreased (since the magnetizing current equals no-load current at no-load operation), the leakage inductance is increased, and the starting current is reduced (Table 5). The disadvantage of the closed rotor slots is the reduction of the motor overload capacity. This is also the case with M1 and M2 models, which have the decreased maximum torque compared with BM model (Table 4).

The influence of the variation of different design parameters on overall motor efficiency is presented in Figure 10. Diagrams are obtained by keeping the starting motor configuration and by varying only one parameter (i.e., Figure 10(a), is variable, =130, L=35 mm, =65.4 mm, RSO=1.5 mm). In Figure 10(b) is variable while all the other parameters remain the same as in the starting model, BM. On the same principle, the other results in Figure 10 are obtained. The efficiency increases with the increase of the conductors in the main winding, the decrease of the main winding current, and the decrease of the copper losses.

Motor output power (P2) is proportional to product D2L where D is the stator bore diameter and L is the core length [15]. Consequently, the increase of motor length gives the increase of the motor output power and the increase of the motor efficiency (η) since output and input power of the motor are related to (22). This type of the motor has increased current magnitude with backward time shift and the increased torque pulsations as the motor approaches the rated speed. To reduce this, the impedance in the auxiliary winding should be reduced and consequently the losses are minimized; i.e., number of conductors in auxiliary winding is decreased (Figure 10(b)).

4.2. FEM Analysis

Parametric analysis serves as a starting point for deriving motor FEM model for verification of proposed models (M1 and M2) in terms of magnetic flux density distribution. Figure 11 presents the distribution of the magnetic flux density for all motor models.

There are some general recommendations for the flux density in the specific parts of the machine. For the stator and the rotor teeth, it should be less than 1.8 T, and for the core (yoke) between 1.3 and 1.5 T [15]. Following these general recommendations, the flux density distribution is within recommended ranges for BM and M1. As for the second optimized model M2, bigger values of the flux density can be observed in the rotor teeth due to the motor construction with closed rotor slots. Yet, this value of the flux density is still below the point of the core saturation with respect to the built-in core material as the saturation point is approximately at 2T (Figure 3). FEM models of the motors are verified in terms of their accuracy by calculating the output torque for one fixed speed. It should be noted that presented diagrams of the torque in Figure 12 do not represent the motor transient characteristics usually plotted during motor acceleration from zero up to the rated speed. The presented diagrams are calculating motor torque at one constant speed, i.e., rated speed, for each of the motor models, within the whole interval. Table 6 presents the comparison between the torque values from the parametric analysis and from the FEM models at rated load operating mode. As the torque in the FEM models has pronounced oscillations, it is calculated as an average value within the time interval from 175 to 200 ms. The presented results in Table 6 and their similarity verify that the FEM models are sufficiently accurate. Therefore, the presented results of the flux density distribution are considered reliable and contribute to the overall estimation of the motor design and the performance.


Output torque (Nm)Parametric analysisFEM analysis

BM0.41350.4271
M10.41780.4293
M20.4180.4435

5. Conclusion

Single-phase capacitor motors are large electricity consumers if their relative high number in various electrical household appliances is taken into consideration. Therefore, the slight improvement of their efficiency contributes to the overall energy efficiency. The paper presents the impact of several motor parameters such as number of conductors in main and auxiliary winding, machine length, air gap length, and opening of the rotor slots on the efficiency of the permanently split capacitor motor. Parametric analysis was run in Maxwell software which resulted in two improved motor models, M1 and M2, with increased efficiency of 7.4% and 10%, respectively, compared with the starting motor model. The difference between models M1 and M2 is the width of rotor slot opening, with the latter having completely closed rotor slots. The models obtained from the parametric analysis were verified with FEM in terms of the magnetic flux density distribution. The results from the FEM analysis proved that model M1 is well designed with respect to the magnetic flux density distribution as the flux density in machine cross-section is within the recommended intervals for the various parts of the machine. Model M2 has the highest value of flux density in rotor teeth although this value is still below the point of core saturation for the built-in magnetic material. The accuracy of the derived FEM models is verified with the similarity of the obtained results of the output torque from FEM and the parametric analysis. By using the latest advancements in the designing programs for the electrical machines, different motor models can be derived in relatively short computing time, resulting in the best possible motor design in terms of predefined objective function. Alterations in the motor design are easily inputted; the results are quickly obtained giving motor’s designers freedom to explore the different motor variants and to find the best design for a certain application. This paper provides a comprehensible overview of the design parameters of the single-phase permanent split capacitor motor and their impact on the motor efficiency. In order to have comparable results, all motor models are derived with the same capacitor as it affects the motor performance. Authors’ further research will be focused on finding the optimal capacitor value for providing the best starting and operating conditions of the motor.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work was supported by the project “Contributions in Mathematical Theory, Mathematical Modeling and Their Application”, financed by “Goce Delcev” University.

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Copyright © 2019 Vasilija Sarac and Tatjana Atanasova-Pacemska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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