# Characteristics Analysis of Single Phase Induction Motor via Equivalent Circuit Method and Considering Saturation Factor

• 투고 : 2013.07.08
• 심사 : 2013.09.25
• 발행 : 2014.01.01

#### 초록

This paper presents a motor characteristics analysis method using an equivalent circuit. Motor characteristics analysis via equivalent circuit is very important for designing a high efficiency single phase induction motor. The accuracy of the motor characteristics depends on the accuracy of the parameters, especially saturation factor, which determines the cyclical relationship in the analysis process. Therefore, using the proposed method, the saturation factor was calculated using the iteration routine and numerical technique. The proposed method was verified by comparing the finite element method results and the dynamo test results of manufactured prototype model.

# 1. Introduction

An electric motor uses above 50% electrical energy. Among the existing electric motors, the induction motor is most widely used. Therefore, a high efficiency induction motor has to be developed to overcome the current energy shortage and to meet the requirements of policies like the Minimum Energy Performance Standard . For designing a high efficiency single phase induction motor, it is important to analyze the characteristics of the motor using an equivalent circuit, to exclude heuristic knowledge like the output coefficient. The accuracy of the characteristics as analyzed using an equivalent circuit depends on the accuracy of the equivalent circuit’s parameters. In particular, the magnetization reactance, which is calculated based on the saturation factor, is very important. If the saturation factor is assumed to have a conventional heuristic value, many errors occur in the characteristics analysis results, and the designed motor efficiency will be reduced.

In this paper, a motor characteristics analysis method using an equivalent circuit is presented. The key strength of this method is that it can be calculated the saturation factor more accurately.

# 2. Equivalent Circuit of Single Phase Induction Motor

In the case of the single phase induction motor, the main winding and the auxiliary winding are not the same. Thus, it has the structure of an unbalanced two phase motor. To determine the characteristics of the motor under unbalanced conditions, a symmetrical two phase equivalent circuit was constructed [2, 3, 4]. Fig. 1 shows the equivalent circuit as having parameters compensated to the main winding part . Fig. 1.Equivalent circuit for the single-phase induction motor with run-capacitor

The parameters of the equivalent circuit can be calculated based on the motor dimensions, material information and electrical specifications, but magnetization reactance is affected by the saturation factor. The magnetization reactance can be expressed as the following Eq. .

μ0 : permeability in free spaceKs : saturation factorKc : Carter coefficientKFe : stacking factor of coreg : airgap lengthLstk : stack lengthτp : pole pitchWm : number of turns for main windingKwm : winding factor of main windingf1 : stator frequencyp1 : pole pair

As the accuracy of the motor characteristics depends on the accuracy of the saturation factor, it is very important to determine the saturation factor accurately.

# 3. Saturation factor

The saturation factor is defined as the following:

Fg : magnetomotive force in airgapFts,tr : magnetomotive force in teeth of stator, rotorFcs,cr : magnetomotive force in yoke of stator, rotor

To calculate the saturation factor, the magnetomotive force in each part of the motor must be determined. For this step, the magnetic circuit method was used. The total magnetomotive force of the induction motor can be expressed as the following Eq. .

I0 : maximum current value at no load

To calculate the magnetomotive force in each part of the motor, the airgap magnetic flux density should be considered. Unlike the three phase induction motor, in the case of the single phase induction motor, the phase difference between the main winding current and the auxiliary winding current should be considered for the airgap magnetic flux density. The airgap magnetic flux density equation can be expressed as following:

Fm1 : magnetomotive force of main windingFa1 : magnetomotive force of auxiliary windingγ : phase difference between main winding current and auxiliary winding currentθes : electrical angle of stator positionω1 : stator angular speed

When using the equivalent circuit, the phase difference should be considered. Thus, the saturation factor required again. Thus, the saturation factor determined the cyclical relationship in the process. Based on this cyclical relationship, the iteration routine and the numerical technique were applied.

# 4. Iteration Routine and Numerical Technique

Fig. 2 shows the iteration routine for the more accurate calculation of the saturation factor. For the basic designed model, the iteration routine starts the calculation process with an initial saturation factor value. After the analysis of the motor characteristics through the equivalent circuit, the airgap magnetic flux density is recalculated with the main and auxiliary winding currents and the phase difference between them. Using the recalculated airgap magnetic flux density, N+1 step value of the saturation factor is computed. The convergence condition of the iteration routine is that the error between the N+1 step value and the N step value is lower than the criterion. Fig. 2.Flow chart of the iteration routine for considering the saturation factor

In the process, the magnetomotive force in each part can be obtained through the magnetic circuit method. Fig. 3 shows the magnetic circuit and magnetic flux path of the induction motor. Fig. 3.Magnetic circuit and magnetic flux path

Using the magnetic circuit method, the total magnetomotive force can be calculated as the following:

The magnetomotive force equation in the airgap can also be calculated as the following:

Bg1,mag : fundamental element of airgap flux density

The magnetomotive force in the airgap can be calculated based on the magnitude of the airgap magnetic flux density, which the iteration routine converged with the initial saturation factor value.

As the magnetic field strength depends on the magnetic characteristics of the iron material and the magnetic flux density, the magnetomotive force calculation in the yoke and teeth of the core requires the H-B curve characteristics of the iron material. In this paper, H-B curve of iron was fitted using a numerical technique, such as the Gaussian 4th fitting function. Using trust region reflective Newton algorithm , the Gaussian 4th fitting function was fitted as the following:

𝑎1 = 1.189ℯ + 17, 𝑎2 = 1.725ℯ + 4, 𝑎3 = 467.4, 𝑎4 = 2.987ℯ + 4𝑏1 = 2.9, 𝑏2 = 2.135, 𝑏3 = 1.838, 𝑏4 = 2.251𝑐1 = 0.1613, 𝑐2 = 0.1535, 𝑐3 = 0.0263, 𝑐4 = 0.4038

This fitting function fit the H-B curve better compared to the conventional fitting function model [7, 8]. Fig. 4 compares the original H-B curve points and the Gaussian 4th fitting function. Fig. 4.Original H-B curve points and the fitted points using Gaussian 4th fitting function

The magnetic flux densities in yoke and teeth of the core can be expressed as the following:

hcs,cr : yoke width of stator, rotor

bts,tr : teeth width of stator, rotor

Using Eq. (7), the magnetic field strength in each part can be calculated from the precalculated magnetic flux densities in the airgap, yoke, and teeth.

The magnetomotive forces in the yoke and teeth of the core can be calculated as the following:

lcs,cr : average flux path length in yoke of stator, rotorHcs,cr : magnetic field strength in yoke of stator, rotor

lts,tr : teeth height of stator, rotorHts,tr : magnetic field strength in teeth of stator, rotor

# 5. Comparison Between Proposed Method and Finite Element Method

Table 1 shows the design specifications, constraints, and heuristic knowledge of the single phase induction motor. The analysis results using the proposed method were compared with FEM results of the designed model. Fig. 5 shows the FEM model and the external circuit with run capacitor [9-11]. Due to the designed motor type was capacitor start and run, the FEM model was connected with the external circuit, including run capacitor. Using the proposed method and FEM analysis, the airgap magnetic flux density was calculated. Fig. 6 shows the airgap magnetic flux density. The error between the magnetic flux density results was less than 10%. Table 2 compares the results. Except for the rotor loss, the results were very close.

Table 1.Design specifications, constraints, and heuristic knowledge Table 2.Analysis results using the proposed method and finite element method  Fig. 5.The finite element model and the external circuit with run-capacitor Fig. 6.Airgap magnetic flux densities from the finite element method and the proposed method

# 6. Manufactured Model and Test Results

The prototype model was manufactured to compare the results of the proposed method. The rotor of the prototype model had aluminum die-casting rotor bars. Fig. 7 shows the prototype model. Fig. 7.Manufactured prototype model

As no standards have been established for the loss separation test method for the single phase induction motor, the input power, output power, and efficiency were measured using a dynamo test set and a power analyzer, under the state of saturated temperature.

Table 3 shows the comparison. There seems to be a little difference between the measurements and the analysis results using the proposed method. It is for this reason that the motor test condition was set to meet the equal output power for efficiency comparison. The efficiency of the prototype motor was determined to be 6% lower than the value obtained through the proposed method. It is thought that self-cooling fan loss, centrifugal force switch that is a device adjoined to the motor’s rotor, automatically changes from start capacitor to run capacitor and the manufacturing error were reasons of the efficiency error. It caused a slip difference at operating condition had the same motor output. The slip difference reduced the measured efficiency of the manufactured model.

Table 3.Comparison of results using the proposed method and dynamo test results of the manufactured motor As a result, it seems that the proposed method result and the tested result of the manufactured model had a slight error. Despite those errors, the proposed method can be said to be precise and effective for designing a high efficiency motor.

# 7. Conclusion

In this paper, characteristics analysis method by equivalent circuit considering the saturation factor was proposed. In order to calculate a more accurate saturation factor, iteration routine and numerical techniques were applied. By comparing with FEM results of designed model and dynamo test results of prototype model, it can be stated that the proposed method can guarantee more accurate analysis results and design output. And the proposed characteristics analysis method can be applied to basic design process in motors like permanent magnet motor.

#### 과제정보

연구 과제 주관 기관 : National Research Foundation of Korea (NRF)

#### 참고문헌

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