A Hybrid Methodology for Analyzing the Performance of Induction Motors with Efficiency Improvement by Specific Commercial Measures

: This paper presents a hybrid methodology to analyze the commercial measures of changing stator windings and adjusting air gap length to upgrade e ﬃ ciency of typical three-phase direct-on-line induction motors with die-cast copper rotor (DCR). The calculation is carried out through combining the time-stepping and time-harmonic ﬁnite element analysis (FEA) and the circuit equivalent circuit model. Typical full-load performance of stator windings with di ﬀ erent air gap lengths are computed by MATLAB invoking the 2D transient and eddy current ﬁeld analysis in ANSYS / MAXWELL. Then, MATLAB scripts about post-processing of the FEA results are used to obtain the full-load running performance including the loss distribution and circulating current. The MATLAB scripts of circuit model built based on the FEA results is used to compare the overload and starting performance. After that, four stators with the four windings and three DCRs with di ﬀ erent air gap of an 11 kW motor are fabricated and tested to validate the calculations. By comparing results from both calculations and measurements, it is shown that the factors of stator windings and air gap length can e ﬀ ectively improve the e ﬃ ciency of the 11 kW DCR induction motor without the addition of extra materials. The motor with the 11 / 12 pitch Y- ∆ series winding and 0.6 mm air gap has the best performance, in terms of e ﬃ ciency, overloading capability and starting performance. Its e ﬃ ciency can increase from 90% to the highest 92.35% by sole adjustment of stator winding and air gap length.


Introduction
Energy issues have caused motor designers, manufacturers and users to pay more attention to efficiency when applying electric motors to production and daily life, as they consume roughly two-thirds of all the electrical energy used by industrial or commercial applications in industrialized countries [1]. Three-phase squirrel-cage induction motors are one of the most widely used motor types in various fields [2][3][4], therefore it is extremely meaningful to increase the efficiency of this kind of motor. models are used in this paper because the commercial measures concerned, such as changing the type of stator windings and the air gap length, have no important influence on end-winding field. If some other measures that change the end-winding field were to be implemented, 3D models should be used instead of 2D ones. Figure 1 shows the detailed structure of the calculation method. The calculation is realized based on the commercial software MATLAB and ANSYS/MAXWELL. Specifically, a MATLAB program is written to assign values to the dimension of an induction motor and the settings in ANSYS/MAXWELL. Therefore, it is convenient to change the parameters of the induction motor. Then the program invokes ANSYS/MAXWELL to build 2D finite element model of the induction motor. After that, the MATLAB program activates the 2D transient field analysis to conduct time-stepping finite element analysis (FEA) at no-load and full-load condition. Then, the results of time-stepping FEA will be processed to obtain the results of no-load and full-load fluxes, currents and losses by the MATLAB program. Based on the full-load results, the MATLAB will invoke the eddy current field analysis to conduct the full-load FEA to get the fundamental losses. Finally, some analytical formulas and the FEA results are used to calculate the leakage reactance in order to analyze the starting and overload performance of induction motors through the MATLAB program. It should be noticed that the MATLAB program also conduct post-processing, such as FFT analysis and mathematical operations, to obtain power factor, efficiency, harmonic current and so on. In order to be more understandable about the hybrid methodology, a substantial portion of the MATLAB codes for invoking and dealing with the no-load simulation are presented in the Appendix A.

Full-Load Performance Calculation
Since the main purpose of the hybrid methodology is to analyze the efficiency of commercial measures on full-load condition, loss calculation is an important part in the calculation. The eddy current loss of rotor end-rings is assumed to be same to simplify the loss calculation. The total losses P l Load of the induction motor at full-load condition can be defined by:

Full-Load Performance Calculation
Since the main purpose of the hybrid methodology is to analyze the efficiency of commercial measures on full-load condition, loss calculation is an important part in the calculation. The eddy current loss of rotor end-rings is assumed to be same to simplify the loss calculation. The total losses P Load l of the induction motor at full-load condition can be defined by:  (1) where P Load 1copper is the copper loss of the stator windings under full-load conditions, P NoLoad iron is the no-load core loss, P Load 2cageF is the fundamental eddy current loss of the rotor squirrel cage under full-load, P Load s is the full-load stray loss and P m is the mechanical loss. The mechanical loss P m can be calculated by an empirical equation [21].
In the calculation, the mechanical loss P m is calculated based on the steady-state speed under full-load conditions. The electromagnetic losses consist of P Load 1copper , P NoLoad iron , P Load 2cageF and P Load s , colored in Figure 2. where P 1copper Load is the copper loss of the stator windings under full-load conditions, P iron NoLoad is the noload core loss, P 2cageF Load is the fundamental eddy current loss of the rotor squirrel cage under full-load, P s Load is the full-load stray loss and P m is the mechanical loss. The mechanical loss P m can be calculated by an empirical equation [21].
In the calculation, the mechanical loss P m is calculated based on the steady-state speed under full-load conditions. The electromagnetic losses consist of P 1copper Load , P iron NoLoad , P 2cageF Load and P s Load , colored in Figure 2.   Figure 3a shows the 2D model and its mesh. The half model is used because of electromagnetic symmetry, and its total number of mesh elements is roughly 16,000 whose results are almost the same as the results of the roughly 26,000 elements obtained by mesh refinement.  The grade of electrical steel laminations used in the induction motor is DW470-50 whose curve of flux density B versus magnetic field strength H is shown in Figure 3b. The calculation process is as described below.
Firstly, the no-load and full-load performance of the induction motor is computed by the classical field-circuit coupled 2D time-stepping FEM coupled with the primary voltage equation, in which the resistance and end-winding leakage inductance of the stator winding and rotor cage are calculated by   Figure 3a shows the 2D model and its mesh. The half model is used because of electromagnetic symmetry, and its total number of mesh elements is roughly 16,000 whose results are almost the same as the results of the roughly 26,000 elements obtained by mesh refinement. where P 1copper Load is the copper loss of the stator windings under full-load conditions, P iron NoLoad is the noload core loss, P 2cageF Load is the fundamental eddy current loss of the rotor squirrel cage under full-load, P s Load is the full-load stray loss and P m is the mechanical loss. The mechanical loss P m can be calculated by an empirical equation [21].
In the calculation, the mechanical loss P m is calculated based on the steady-state speed under full-load conditions. The electromagnetic losses consist of P 1copper Load , P iron NoLoad , P 2cageF Load and P s Load , colored in Figure 2.   Figure 3a shows the 2D model and its mesh. The half model is used because of electromagnetic symmetry, and its total number of mesh elements is roughly 16,000 whose results are almost the same as the results of the roughly 26,000 elements obtained by mesh refinement.  The grade of electrical steel laminations used in the induction motor is DW470-50 whose curve of flux density B versus magnetic field strength H is shown in Figure 3b. The calculation process is as described below.
Firstly, the no-load and full-load performance of the induction motor is computed by the classical field-circuit coupled 2D time-stepping FEM coupled with the primary voltage equation, in which the resistance and end-winding leakage inductance of the stator winding and rotor cage are calculated by The grade of electrical steel laminations used in the induction motor is DW470-50 whose curve of flux density B versus magnetic field strength H is shown in Figure 3b. The calculation process is as described below. Firstly, the no-load and full-load performance of the induction motor is computed by the classical field-circuit coupled 2D time-stepping FEM coupled with the primary voltage equation, in which the resistance and end-winding leakage inductance of the stator winding and rotor cage are calculated by the analytical formulas [21]. For a 2D XY problem, the vectors have only one component in the z-direction. The 2D time-dependent magnetic equation [29] of an induction motor is expressed as: where υ is the reluctivity, A is the magnetic vector potential, σ is the conductivity, t is the time, J s is the source current density that is given by [29]: where d f is the polarity (+1 or −1) to represent forward or return paths of stator windings, N f is the total conductor number of the filaments in the winding, S f is the total area of the cross-section of the region occupied by the winding, a is the number of parallel branches in the winding, p is the ratio of the original full model to the field domain to be solved, i f is the total terminal current flowing into a filament winding which is calculated by: where L is the axial length, R dc is the total DC resistance of the straight portion of the stator windings, R end is the end-turn resistance of the stator windings, L end is the end-turn inductance of the stator windings, u s is the source voltage of the stator windings, Ω is the cross-section of the conductors. The magnetic vector potential A and current is obtained through Equations (2)-(4), then other parameters, such as the magnetic flux, electromagnetic torque, etc., are calculated. The input of the time-stepping FEM is the voltage source of stator windings and the rotor rotating speed. The no-load speed is set at the synchronous speed n s (unit: rpm). The full-load simulation proceeds as follows. Note that searching the input of the full-load speed is realized by a MATLAB script with the secant method: (1) Assuming rotating speeds n 1 and n 2 (n 1 n 2 , unit: rpm) that are a little less than the synchronous speed as the input speeds, calculate their output torques T 1 and T 2 by the time-stepping FEM and the empirical equation [21], respectively. Note that output torque T j is calculated by: In Equation (5), p denotes the number of pole pairs, D 1out denotes the stator outer diameter and its unit is m, T em denotes the electromagnetic torque computed by the time-stepping FEM. If: then the speed fulfilling Equation (6) is the full-load speed and the algorithm goes to Step 4; otherwise, subscript i = 3 and the algorithm goes to Step 2. In Equation (6), T n denotes the full-load torque 70.8 Nm corresponding to 11 kW output power and ε denotes the allowable error. (2) Calculate the speed n i by: then n i is the full-load speed and the algorithm proceeds to Step 4; otherwise i = i + 1 and the algorithm goes to Step 2. (4) Using the full-load speed calculate the full-load results. By solving the electromagnetic equations coupled to the primary voltage equation, they are obtained that the stator current I 1m at each phase m (m = A, B, C.), and the current density of the rotor bar J r . Then the copper loss of stator windings P Load 1copper and the total eddy current loss of the rotor squirrel cage P Load 2cage on full-load condition can be calculated. The full-load core loss P Load iron and no-load core loss P NoLoad iron are computed based on the magnetic flux.
Then the classical 2D time-harmonic FEA is applied to compute the fundamental eddy current loss of the rotor squirrel cage P Load 2cageF . Similarly, the phasors in 2D time-harmonic FEA have only one component in the z-direction. Its equation is: where ω is the fundamental frequency, I is the stator winding current. The input of the 2D time-harmonic FEA is the fundamental component of FFT analysis of stator winding current from the 2D time-stepping FEM results. Note that the rotor bar conductivity should be multiplied by full-load slip obtained from the 2D time-stepping FEM results rather than setting the rotor rotating speed. The fundamental eddy current loss of the rotor squirrel cage P Load 2cageF can be calculated based on the results of the fundamental current density of rotor bar J rF .
According to the definition in the IEEE standard 112-method B [30], the stray loss is calculated by: Finally, all the losses colored in Figure 2 are calculated through the FEA of ANSYS/MAXWELL and its results processed by MATLAB. The efficiency η can be obtained by losses and output power. Power factor cos φ is calculated based on the fundamental angle between the phase voltage and the phase current which is computed by FFT analysis. Additionally, circulating currents in ∆ windings can be separated from the fundamental currents.

Overload and Starting Performance Calculation
The overload and starting performance is evaluated based on the T equivalent circuit model. The calculation is shown in the part of "Analytical Formulas" in Figure 1.
The overload performance of an induction motor is determined by its peak torque T max .
Since R 2 1 (X σ1 +X σ2 2 , the peak torque T max can approximately be: where U Nϕ is the rated phase voltage, R 1 is the phase resistance of the stator winding, X σ1 is the total steady-state stator leakage reactance, X σ2 is the total steady-state rotor cage leakage reactance (reported to the stator). Equation (12) shows the peak torque of induction motors is inversely proportional to the total stator and rotor leakage reactances.
Because the induction motor studied in this paper is direct-on-line starting, it is necessary to analyze the starting performance. The starting torque T st can be calculated by: where R 2(st) is the rotor cage resistance (reported to the stator) at starting stage and Z st is the magnitude of total impedance at start. It should be noticed R 2(st) is calculated based on skin effect [21]. During starting stage, magnetizing current can be neglected because the voltage drop of stator impedance is much higher than those at the steady-state stage. Therefore, the total starting impedance Z st can be calculated approximately by the T model: where R 1 is the phase resistance of the stator winding, X σ1(st) is the total stator leakage reactance at starting stage and X σ2(st) is the total rotor leakage reactance at start (reported to the stator).

Specific Commercial Measures for Efficiency Improvement
This paper investigates commercial measures, such as changing the type of stator windings and the air gap length, to further raise the efficiency of an 11 kW three-phase squirrel-cage induction motor with a DCR. The main characteristics are shown in Table 1. It should be noticed that the parameters of die-cast copper rotor are the same as typical cast aluminum rotor of an 11 kW induction motor to reduce manufacturing cost. The distinction between two of them comes from the fact that its rotor squirrel cage has been changed to copper and its outer diameter is adjusted for different air gap lengths. Two types of stator windings, namely ∆ winding and Y-∆ series winding, are studied. The ∆ winding is the single-double-layer (S.D.L.) winding. As shown in Figure 4a, the single-double-layer ∆ winding, which evolves from the conventional double-layer winding Figure 4b, is generally short pitched. Owning to the change, the end-winding of the single-double-layer winding is shorter than the double-layer winding. Thus, the single-double-layer winding has a smaller phase stator resistance than the overlap winding.
Y-∆ series windings, which are formed in outer-Y-inner-∆ hybrid connection, are shown in Figure 5a. Its loop current in the inner-∆ windings is relatively smaller than the outer-∆-inner-Y windings. The basic principles of three-phase ∆-Y series windings are shown: (1) Current density of inner-∆ winding must be equal to the outer-Y. (2) Number of turns in series of inner-∆ winding equal to square root of three times as outer-Y windings.
Stator inner diameter, D1in (mm) 180 Axial length, L (mm) 170 Service S1 Two types of stator windings, namely ∆ winding and Y-∆ series winding, are studied. The ∆ winding is the single-double-layer (S.D.L.) winding. As shown in Figure 4a, the single-double-layer ∆ winding, which evolves from the conventional double-layer winding Figure 4b, is generally short pitched. Owning to the change, the end-winding of the single-double-layer winding is shorter than the double-layer winding. Thus, the single-double-layer winding has a smaller phase stator resistance than the overlap winding.  Y-∆ series windings, which are formed in outer-Y-inner-∆ hybrid connection, are shown in Figure 5a. Its loop current in the inner-∆ windings is relatively smaller than the outer-∆-inner-Y windings. The basic principles of three-phase ∆-Y series windings are shown: (1) Current density of inner-∆ winding must be equal to the outer-Y.
(2) Number of turns in series of inner-∆ winding equal to square root of three times as outer-Y windings. Three typical Y-∆ series windings, namely single-layer Y-∆ series winding, double-layer Y-∆ series winding with a 2/3 pitch and an 11/12 pitch are studied. The winding coefficients of the Y-∆ series windings are higher because of their narrow phase belt. Table 2 shows the specific parameters of the stator windings, where nstr is number of wires per conductor, and dw is the wire diameter, q is the number of slots per pole per phase, α is the electric angle between two stator slots, y is the coil span in terms of stator slots, and τ is pole pitch in terms of stator slots, p is the number of pole pairs, R1sm is the phase resistance of stator windings, N is the number of turns in series per phase. The four windings are numbered to simplify the introduction in Table 2. As the number of turns in series per phase N must be an integer number and the four types of windings are designed to keep the same fundamental magnetomotive force (MMF), N∆/NY of Y-∆ series windings are close to √3 . The cross-sectional area per conductor of the outer-Y winding must be approximately √3 times larger than those of the inner-∆ winding in order to make the current density to be the same. The number of parallel branches of these four stator windings is 1. Three typical Y-∆ series windings, namely single-layer Y-∆ series winding, double-layer Y-∆ series winding with a 2/3 pitch and an 11/12 pitch are studied. The winding coefficients of the Y-∆ series windings are higher because of their narrow phase belt. Table 2 shows the specific parameters of the stator windings, where n str is number of wires per conductor, and d w is the wire diameter, q is the number of slots per pole per phase, α is the electric angle between two stator slots, y is the coil span in terms of stator slots, and τ is pole pitch in terms of stator slots, p is the number of pole pairs, R 1sm is the phase resistance of stator windings, N is the number of turns in series per phase. The four windings are numbered to simplify the introduction in Table 2. As the number of turns in series per phase N must be an integer number and the four types of windings are designed to keep the same fundamental magnetomotive force (MMF), N ∆ /N Y of Y-∆ series windings are close to √ 3. The cross-sectional area per conductor of the outer-Y winding must be approximately √ 3 times larger than those of the inner-∆ winding in order to make the current density to be the same. The number of parallel branches of these four stator windings is 1.

Harmonic Analysis of Stator Windings
The γ-th harmonic amplitude of the MMF for the single-double-layer winding is [18]: γy sd τ sd ) (15) and the γ-th harmonic amplitude of the MMF for the Y-∆ series windings is [18]: In Equations (15)- (18), the definition of N, p, q, y, τ, α is the same as Table 2. γ = 6k ± 1, k = 0, 1, 2, . . . because the (3k)th harmonics counteract in the three-phase windings. The subscript sd denotes the single-double-layer winding, and Y-∆ denotes the Y-∆ series hybrid winding. I is the root-mean-square (RMS) value of the phase-current in the stator winding. Figure 6 shows the MMF distribution of the four types of windings calculated by Equations (15)- (18), where harmonic content is defined by the MMF ratio of the specific harmonic to the fundamental harmonic. The winding #1 contains (6k ± 1) th (k = 1, 2, 3, . . . .) harmonics where the 5th and 7th harmonics are reduced by the 5/6 pitch. From the Equation (16), it can be found that the MMF of the Y-∆ series windings is zero if the MMF of the inner Y winding is equivalent to the outer ∆ winding and k is an odd number. Therefore, the windings #2 and #4 approximately contain (6k ± 1)th harmonics where k = 2, 4, 6, . . . because the harmonic content of k = 1, 3, 5, . . . are nearly suppressed. The winding #3 contains the least harmonics that are approximately (12k ± 1)th harmonics where k = 2, 4, 6, . . .

Air Gap Length
The air gap length should always be small to decrease the no-load current of induction motors. There is mechanical minimum limitation for the air gap length to prevent the possible contact between stator and rotor. Reference [21] offers the following equation for estimation of the air gap length g: (19) where the unit is meters, D1in is the stator inner diameter, L is the axial length.
The stator differential leakage reactance Xd1 can be calculated by [21]: where μ0 is the magnetic constant, τ is the pole pitch in terms of length, kc is the Carter coefficient, ksat is the saturation factor, Fδ is the air gap MMF drop, Ft1 is the stator tooth MMF drop, Ft2 is the rotor tooth MMF drop, and kwγ is the winding factor of the γth harmonic.
Equation (20) shows that harmonic leakage reactance is inversely proportional to air gap length. Therefore, air gap should be somewhat large enough to make sure harmonic leakage reactance is small so as its stray loss. According to Equation (19), air gap g of the 11 kW induction motor is approximately 0.5 mm. Therefore, the 11 kW induction motor with air gap varying around 0.5 mm is analyzed.

Calculation Results and Evaluation
Through the calculation above, the full-load, overload and starting performance of the 11 kW induction motor with a DCR are computed and compared with each other for the commercial measures that changing types of stator windings and air gap length.

Air Gap Length
The air gap length should always be small to decrease the no-load current of induction motors. There is mechanical minimum limitation for the air gap length to prevent the possible contact between stator and rotor. Reference [21] offers the following equation for estimation of the air gap length g: where the unit is meters, D 1in is the stator inner diameter, L is the axial length.
The stator differential leakage reactance X d1 can be calculated by [21]: where µ 0 is the magnetic constant, τ is the pole pitch in terms of length, k c is the Carter coefficient, k sat is the saturation factor, F δ is the air gap MMF drop, F t1 is the stator tooth MMF drop, F t2 is the rotor tooth MMF drop, and k wγ is the winding factor of the γth harmonic. Equation (20) shows that harmonic leakage reactance is inversely proportional to air gap length. Therefore, air gap should be somewhat large enough to make sure harmonic leakage reactance is small so as its stray loss. According to Equation (19), air gap g of the 11 kW induction motor is approximately 0.5 mm. Therefore, the 11 kW induction motor with air gap varying around 0.5 mm is analyzed.

Calculation Results and Evaluation
Through the calculation above, the full-load, overload and starting performance of the 11 kW induction motor with a DCR are computed and compared with each other for the commercial measures that changing types of stator windings and air gap length. preliminarily thermal analysis which is not presented in this paper. The no-load condition is set at the synchronous speed of 1500 rpm. The no-load temperature is assumed as the room temperature of 15 • C. preliminarily thermal analysis which is not presented in this paper. The no-load condition is set at the synchronous speed of 1500 rpm. The no-load temperature is assumed as the room temperature of 15 °C.   preliminarily thermal analysis which is not presented in this paper. The no-load condition is set at the synchronous speed of 1500 rpm. The no-load temperature is assumed as the room temperature of 15 °C.             winding #1 winding #2 winding #3 winding #4 Figure 11. Calculation results of power factor versus air gap length for the four stator windings.             Figure 7 shows the output power of the 11 kW induction motors with the four types of stator windings with various air gap lengths. Since the rotating speed of the motors decrease slightly with the increased air gap lengths for each stator winding, the maximum reduction of the output power of the motor with each stator winding is less than 4 W by increasing the air gap length. Therefore, the output powers for the motors with stator winding #1, #2, #3 and #4 are approximately 10.98, 11.02, 11.04 and 11.03 kW. Figure 8 compares the losses of the induction motor with stator windings and various air gap lengths. As shown, the mechanical losses of the motor with the four windings are almost constant because the speed for each stator winding is roughly constant. The full-load fundamental rotor cage losses and no-load core losses reduce slightly with the increased air gap lengths. However, the full-load copper loss and stray loss change greatly with the increased air gap lengths. As seen in Figure 9, the line current of the motor raises with the increased air gap lengths so as copper losses. The copper losses of the winding #1 and #4 are greater than those in #2 and #3 because the fundamental winding factors of the #1 and #4 are relatively lesser. Although the proportion of the stray loss is not high in the total losses, it decreases significantly with the increased air gap lengths. As a consequence of low harmonics, the stray loss of the #3 winding is least in the four types of windings regardless of any air gap length. Furthermore, the total losses of the motor with the four windings reach the minimum value with air gap is set between 0.5 mm and 0.6 mm. Based on the results, the efficiency of the windings can be obtained as shown in Figure 10. Obviously, the efficiency of the #3 is the highest among all these windings. The efficiency of #2 is only slightly lower than the #3, while #4 is the lowest. Besides, when air gap length is 0.6 mm, the efficiencies of #1, #2 and #3 reach highest while #4 can reach highest when its air gap length is set as 0.5 mm. Figure 11 shows the power factors of the motors. As seen, the power factors of the motors with the four types of windings reduce with the increased air gap lengths. In particular, power factor of the winding #3 is lowest and the power factors of #4, #2 and #1 all increase accordingly.

Full-Load Performance
From the calculated results, it can be shown that winding #3 has the highest efficiency because it owns the second highest winding factor with the lowest harmonics. Superficially, it seems that the winding #2 is better than the #3 because the efficiency of the #2 is close to the #3 while its power factor is a little bit higher than that in #3. In order to keep the fundamental MMF same as the other three windings, the ratio of the number of turns for the winding #3 in series per phase of the inner ∆ to the outer Y, N ∆ /N Y , is 1.667. This particular value is not fairly close to √ 3 so that the electromagnetic fields of the inner ∆ and the outer Y winding are distorted. Therefore, the power factor and the current of the winding #3 are both deteriorated. If the ratio of the winding #3 is close to √ 3, as similar as its counterpart does, i.e., #2 and #3 are 1.750 and 1.714, respectively, its power factor and current could be close to that in #2. Furthermore, the efficiency of the #3 winding would be higher because of it consists of smaller copper loss.
Although the suggested value by empirical equation is 0.5 mm, it should also be noted the best air gap lengths of the efficiency should instead be 0.6 mm for the #1, #2 and #3 and 0.5mm for the #4. The difference is caused possibly by the greater copper loss in the #4 winding. The relatively smaller air gap in #4 winding results with less current so as copper loss, and hence resulting the highest efficiency.
Besides (3k)th (k = 1, 2, 3, . . . .) harmonics cause circulating zero-order currents in ∆ windings and it will increase copper loss. The steady-state full-load phase currents of the ∆ winding in the four stator windings with different air gap lengths can be calculated by 2D FEA. Figure 12 shows the steady-state phase currents in the ∆ winding of the four stator windings with 0.3 mm air gap. It can be seen that the current waveforms are distorted because of harmonic current. The waveform distortion of the winding #2 and #3 is more severe than those in winding #1 and #4. The results imply the winding #2 and #3 contain more current harmonics. The 3rd harmonic current amplitude I h3 is analyzed by using FFT because it is the main harmonic content that causes circulating zero-order current whose results are shown in Figure 13. Then, the stator copper losses caused by 3rd harmonic current are calculated and shown in Figure 14. Since greater air gap reduces harmonic amplitude, all 3rd harmonic currents of the four windings decrease with the increased air gap lengths. Furthermore, the 3rd harmonic current of the winding #2 is close to the winding #3 and they are much higher than those in winding #1 and #4. The winding #4 contains the lowest 3rd harmonic current because of its 2/3 pitch eliminates (3k)th (k = 1, 2, 3, . . . .) harmonics. Therefore, copper loss caused by 3rd harmonic current of winding #4 is close to zero. It is the lowest value among all four windings, although it is also the Y-∆ series winding. The copper loss of 3rd harmonic current of the winding #1 is a litter higher than the winding #4, while it is much less than those in winding #2 and #3. They are both Y-∆ series winding whose inner ∆ winding has remarkable 3rd circulating current.

Overload Performance
In the Equation (12), the total stator and the total rotor leakage reactances are calculated by: where X s1 is stator slot leakage reactance, X d1 is starting stator differential leakage reactance and X e1 is starting stator end-winding leakage reactance, X s2 is the rotor slot leakage reactance (reported to the stator), X d2 is the rotor differential leakage reactance (reported to the stator) and X e2 is the rotor end-winding leakage reactance (reported to the stator). Equation (20) shows that the stator differential leakage reactance X d1 is proportional to ∞ γ=2 (k γ / γ) 2 which is the differential leakage coefficient. The calculated values of the differential leakage coefficients of single-double layer ∆ winding with q = 4, and Y-∆ series winding with q = 2 for various pitches are shown in Figure 15. It can be seen that the coefficients of Y-∆ series winding are much less than those in single-double layer ∆ winding. It is because (6k ± 1)th harmonics, where k is odd number, are suppressed in the Y-∆ winding. those in winding #1 and #4. The winding #4 contains the lowest 3rd harmonic current because of its 2/3 pitch eliminates (3k)th (k = 1, 2, 3, ….) harmonics. Therefore, copper loss caused by 3rd harmonic current of winding #4 is close to zero. It is the lowest value among all four windings, although it is also the Y-∆ series winding. The copper loss of 3rd harmonic current of the winding #1 is a litter higher than the winding #4, while it is much less than those in winding #2 and #3. They are both Y-∆ series winding whose inner ∆ winding has remarkable 3rd circulating current.

Overload Performance
In the Equation (12), the total stator and the total rotor leakage reactances are calculated by: where X s1 is stator slot leakage reactance, X d1 is starting stator differential leakage reactance and X e1 is starting stator end-winding leakage reactance, X s2 ' is the rotor slot leakage reactance (reported to the stator), X d2 ' is the rotor differential leakage reactance (reported to the stator) and X e2 ' is the rotor end-winding leakage reactance (reported to the stator). Equation (20) shows that the stator differential leakage reactance X d1 is proportional to which is the differential leakage coefficient. The calculated values of the differential leakage coefficients of single-double layer ∆ winding with q = 4, and Y-∆ series winding with q = 2 for various pitches are shown in Figure 15. It can be seen that the coefficients of Y-∆ series winding are much less than those in single-double layer ∆ winding. It is because (6k ± 1)th harmonics, where k is odd number, are suppressed in the Y-∆ winding.  Figures 16 and 17 show the per unit values of the stator differential leakage reactance x d1 and rotor differential leakage reactance x d2 ' (reported to the stator) of the four stator windings, with air gap varying from 0.3 mm to 0.8 mm. Obviously, the stator and rotor differential leakage reactances decrease with increased air gap. The reason is that the differential leakage reactance is inversely proportional to equivalent air gap length, as described in the Equation (20). Due to minimum number of turns in series per phase and harmonic contents, winding #3 has the lowest differential leakage reactance.  Figures 16 and 17 show the per unit values of the stator differential leakage reactancex d1 and rotor differential leakage reactancex d2 (reported to the stator) of the four stator windings, with air gap varying from 0.3 mm to 0.8 mm. Obviously, the stator and rotor differential leakage reactances decrease with increased air gap. The reason is that the differential leakage reactance is inversely proportional to equivalent air gap length, as described in the Equation (20). Due to minimum number of turns in series per phase and harmonic contents, winding #3 has the lowest differential leakage reactance.
where kw1 is the fundamental winding factor, Nr is the number of rotor slots, s2 is the rotor slot leakage coefficient which is determined by the rotor slot geometry [21] including the rotor slot bridge. As shown in Figure 18, the rotor slot leakage reactance x s2 ' (reported to the stator) also varies with air gap length. Due to the total length of the air gap and rotor slot bridge keeps constant, the rotor slot leakage reactance increases at the beginning and then decrease with increased air gap which causes the length reduction of the rotor slot bridge. The stator and rotor leakage reactance including stator slot leakage reactance x s1 , stator end-winding leakage reactance x e1 and rotor end-winding leakage reactance x e2 ' (reported to the stator), are summarized in Table 3. Furthermore, the reciprocal of total stator and rotor leakage reactances of the 11 kW induction motors with four stator windings, with various air gap lengths are calculated and shown in Figure 19. Regarding any air gap lengths, winding #3 owns the highest value with its peak torque roughly 1.1 times larger than the other three windings. The peak torque of winding #2 is somewhat greater than winding #4, and peak torque of   Figure 17. Steady-state rotor differential leakage reactance versus different air gap for the four stator windings.
The per unit value of rotor slot leakage reactance x s2 ' (reported to the stator) can be calculated by [21]: where kw1 is the fundamental winding factor, Nr is the number of rotor slots, s2 is the rotor slot leakage coefficient which is determined by the rotor slot geometry [21] including the rotor slot bridge. As shown in Figure 18, the rotor slot leakage reactance x s2 ' (reported to the stator) also varies with air gap length. Due to the total length of the air gap and rotor slot bridge keeps constant, the rotor slot leakage reactance increases at the beginning and then decrease with increased air gap which causes the length reduction of the rotor slot bridge. The stator and rotor leakage reactance including stator slot leakage reactance x s1 , stator end-winding leakage reactance x e1 and rotor end-winding leakage reactance x e2 ' (reported to the stator), are summarized in Table 3. Furthermore, the reciprocal of total stator and rotor leakage reactances of the 11 kW induction motors with four stator windings, with various air gap lengths are calculated and shown in Figure 19. Regarding any air gap lengths, winding #3 owns the highest value with its peak torque roughly 1.1 times larger than the other three windings. The peak torque of winding #2 is somewhat greater than winding #4, and peak torque of The per unit value of rotor slot leakage reactancex s2 (reported to the stator) can be calculated by [21] where k w1 is the fundamental winding factor, Nr is the number of rotor slots, λ s2 is the rotor slot leakage coefficient which is determined by the rotor slot geometry [21] including the rotor slot bridge. As shown in Figure 18, the rotor slot leakage reactancex s2 (reported to the stator) also varies with air gap length. Due to the total length of the air gap and rotor slot bridge keeps constant, the rotor slot leakage reactance increases at the beginning and then decrease with increased air gap which causes the length reduction of the rotor slot bridge. The stator and rotor leakage reactance including stator slot leakage reactancex s1 , stator end-winding leakage reactancex e1 and rotor end-winding leakage reactancex e2 (reported to the stator), are summarized in Table 3. Furthermore, the reciprocal of total stator and rotor leakage reactances of the 11 kW induction motors with four stator windings, with various air gap lengths are calculated and shown in Figure 19. Regarding any air gap lengths, winding #3 owns the highest value with its peak torque roughly 1.1 times larger than the other three windings. The peak torque of winding #2 is somewhat greater than winding #4, and peak torque of winding #1 is close to the #2. Therefore, the induction motor with winding #3 consists of the better overload performance than all the other three windings. Additionally, the overload capability can be enhanced by increasing the air gap length. w w w w Figure 18. Steady-state rotor slot leakage reactances versus different air gap for the four stator windings. Table 3. Stator and rotor steady-state leakage reactances of the four stator windings. winding #1 winding #2 winding #3 winding #4 Figure 19. Reciprocal of total leakage reactances versus different air gap for the four stator windings.

Starting Performance
In Equation (14), the total stator and the total rotor leakage reactance at starting stage is calculated by:      winding #1 winding #2 winding #3 winding #4 Figure 19. Reciprocal of total leakage reactances versus different air gap for the four stator windings.

Starting Performance
In Equation (14), the total stator and the total rotor leakage reactance at starting stage is calculated by:

Starting Performance
In Equation (14), the total stator and the total rotor leakage reactance at starting stage is calculated by: where k st is the starting leakage flux saturation coefficient which is obtained by looking up the curve of k st versus the virtual air gap flux density B L for the starting flux leakage [21]. Assuming the stator and rotor slot MMF drop takes place only in the air gap for the starting flux leakage, the virtual air gap flux density is calculated by the stator and rotor slot MMF of the starting flux leakage. Besides, the stator and rotor slot MMF is calculated by the starting stator current I st1 . Note that I st1 is obtained by iterative calculation. Specifically, making an assumption of I st1(0) is to calculate the total starting impedance Z st , then the starting stator current I st1 (1) is computed based on the results of Z st . This iterative calculation continues until the value of |I st1(i+1) − I st1(i) | (i = 0, 1, 2, . . . ) is less than an allowable error. λ s is the slot leakage coefficient, the subscript 1 denotes the stator, the subscript 2 denotes the rotor and the subscript (st) denotes the starting parameter. λ s1 and λ s2 is determined by the stator and the rotor slot geometry respectively. λ s1(st) and λ s2(st) is calculated by the stator and the rotor slot geometry and the coefficient k st .
The results of the coefficient k st of the four stator windings versus air gap length are shown in Figure 20. The starting stator and the rotor differential leakage reactances X d1(st) and X d2(st) can be obtained by the results of the coefficient k st , Figures 16 and 17. The starting stator and rotor slot leakage reactances X s1(st) and X s2(st) are also calculated through the coefficient k st and the steady-stator values. Besides, the stator and rotor end-winding leakage reactances are the same as the steady-state parameters in Table 3.
where kst is the starting leakage flux saturation coefficient which is obtained by looking up the curve of kst versus the virtual air gap flux density BL for the starting flux leakage [21]. Assuming the stator and rotor slot MMF drop takes place only in the air gap for the starting flux leakage, the virtual air gap flux density is calculated by the stator and rotor slot MMF of the starting flux leakage. Besides, the stator and rotor slot MMF is calculated by the starting stator current Ist1. Note that Ist1 is obtained by iterative calculation. Specifically, making an assumption of Ist1(0) is to calculate the total starting impedance Z st , then the starting stator current Ist1 (1) is computed based on the results of Z st . This iterative calculation continues until the value of |Ist1(i+1) − Ist1(i)| (i = 0, 1, 2, …) is less than an allowable error. s is the slot leakage coefficient, the subscript 1 denotes the stator, the subscript 2 denotes the rotor and the subscript (st) denotes the starting parameter. s1 and s2 is determined by the stator and the rotor slot geometry respectively. s1(st) and s2(st) is calculated by the stator and the rotor slot geometry and the coefficient kst.
The results of the coefficient kst of the four stator windings versus air gap length are shown in Figure 20. The starting stator and the rotor differential leakage reactances 1( ) and  Table 3. Assuming the starting rotor resistances (reported to the stator) keep constant for the different stator windings and air gap length, their starting torques only vary with the magnitude of total starting impedance. Thus, the starting torques can be seen to be directly proportional to the reciprocals of magnitude square of total impedance at starting 1/Z st 2 , as described in Equation (13). Figure 21 shows the calculated per unit results of the parameters 1/̂s t 2 of the motors with different stator windings and air gap lengths. It can be seen that they decrease with increased air gap lengths, with reduced amplitudes confined within 5% for all the four windings. Besides, the parameter 1/̂s t 2 of winding #3 is much higher than the other three windings. The parameters 1/̂s t 2 of the#1, #2 and #4 decrease when the #4 is close to the #1 because the #4 winding has much greater stator resistance and maximum number of turns in series per phase. Therefore, the starting performance of the winding #3 is superior to the other three windings with its starting torque is roughly 1.3 times higher than the others. Assuming the starting rotor resistances (reported to the stator) keep constant for the different stator windings and air gap length, their starting torques only vary with the magnitude of total starting impedance. Thus, the starting torques can be seen to be directly proportional to the reciprocals of magnitude square of total impedance at starting 1/Z 2 st , as described in Equation (13). Figure 21 shows the calculated per unit results of the parameters 1/ẑ 2 st of the motors with different stator windings and air gap lengths. It can be seen that they decrease with increased air gap lengths, with reduced amplitudes confined within 5% for all the four windings. Besides, the parameter 1/ẑ 2 st of winding #3 is much higher than the other three windings. The parameters 1/ẑ 2 st of the#1, #2 and #4 decrease when the #4 is close to the #1 because the #4 winding has much greater stator resistance and maximum number of turns in series per phase. Therefore, the starting performance of the winding #3 is superior to the other three windings with its starting torque is roughly 1.3 times higher than the others.

Experimental Results
The 11 kW induction motor with four types of stator windings and three non-skewed die-cast copper rotors corresponding to air gap length of 0.4, 0.6 and 0.8 mm are fabricated and tested to validate the calculation results of the hybrid methodology. These stators and rotors can be assembled into twelve 11 kW prototypes, respectively. No-load and full-load tests of the prototypes are carried out in accordance with the standard of IEEE 112-B [30]. At the full-load test, the shaft torque method is used to obtain data with higher accuracy [31]. Figure 22 shows one of the 11 kW prototypes with its experimental test bench. It can be seen that a high precision torque-speed transducer is installed between the prototype and a dynamometer to make sure the output torque is 70.8 Nm on full-load condition. The experimental devices are summarized in Table 4. The signal of the torque-speed transducer is used to test the output torque and speed of the motor shaft directly. The signal of three current transducers and voltage from the output of voltage transformer are input into the highprecision power analyzer.

Experimental Results
The 11 kW induction motor with four types of stator windings and three non-skewed die-cast copper rotors corresponding to air gap length of 0.4, 0.6 and 0.8 mm are fabricated and tested to validate the calculation results of the hybrid methodology. These stators and rotors can be assembled into twelve 11 kW prototypes, respectively. No-load and full-load tests of the prototypes are carried out in accordance with the standard of IEEE 112-B [30]. At the full-load test, the shaft torque method is used to obtain data with higher accuracy [31]. Figure 22 shows one of the 11 kW prototypes with its experimental test bench. It can be seen that a high precision torque-speed transducer is installed between the prototype and a dynamometer to make sure the output torque is 70.8 Nm on full-load condition. The experimental devices are summarized in Table 4. The signal of the torque-speed transducer is used to test the output torque and speed of the motor shaft directly. The signal of three current transducers and voltage from the output of voltage transformer are input into the high-precision power analyzer.

Experimental Results
The 11 kW induction motor with four types of stator windings and three non-skewed die-cast copper rotors corresponding to air gap length of 0.4, 0.6 and 0.8 mm are fabricated and tested to validate the calculation results of the hybrid methodology. These stators and rotors can be assembled into twelve 11 kW prototypes, respectively. No-load and full-load tests of the prototypes are carried out in accordance with the standard of IEEE 112-B [30]. At the full-load test, the shaft torque method is used to obtain data with higher accuracy [31]. Figure 22 shows one of the 11 kW prototypes with its experimental test bench. It can be seen that a high precision torque-speed transducer is installed between the prototype and a dynamometer to make sure the output torque is 70.8 Nm on full-load condition. The experimental devices are summarized in Table 4. The signal of the torque-speed transducer is used to test the output torque and speed of the motor shaft directly. The signal of three current transducers and voltage from the output of voltage transformer are input into the highprecision power analyzer.    The resistances of the stator windings are measured at the room temperature of 3 • C. The measured data are shown in Table 5. It can be seen that the phase resistances at full-load test are smaller than the calculated values in Table 2. The difference mainly comes from inaccurate calculation of end-winding lengths which is acceptable. Experimental results of efficiency and power factor are shown in Figures 23 and 24, respectively. As shown, with the air gap length of 0.6 mm, winding #3 consists of the highest efficiency. It somehow consists of lower power factor than the others, with differences increasing from 90% to 92.35% by only changing the stator winding and air gap length. Figures 23 and 24 also show the relative errors comparing the experimental and simulation results. The relative error ε is defined as: where x e is the experimental value, x s is the simulation value. It can be seen that the experimental efficiency is close to the simulation results that all the relative errors are less than 2%. The relative errors of the power factor are a little greater than the efficiency, especially for the winding #2, #3 and #4 which are all the Y-∆ series windings. The reason is the imbalance between the outer-Y winding and the inner-∆ in the Y-∆ series windings in the external circuit model of the Y-∆ series windings coupled with 2D FEA model, which causes the current distortion leading the current to lag behind the voltage. That means the power factor is sensitive to the unbalanced end-winding leakage inductances in the external circuit, in which the end-winding leakage inductances of the inner-∆ winding should be equal to three times as much as the outer-Y winding. Although the leakage inductances are calculated by analytical equations in [21], it is difficult to strictly meet this requirement. Therefore, the calculation of the power factor for the winding #2, #3 and #4 is lower than the experimental value. That is why the highest power factor in the experiment is the winding #2 rather than the simulation result of winding #1. Fortunately, the relative errors of the power factor are less than 9% which fulfil the engineering requirement. Furthermore, the influence of the power factor error is relatively small on the computational precision of efficiency. It is expected the efficiency can reach 90% if cast-aluminum rotor is replaced with cast-copper rotor. Obviously the performance of winding #3 can be further improved by adjusting its ratio of the number of turns in series per phase of the inner ∆ to the outer Y-N ∆ /N Y . The weight of these four windings is all around 9 kg which is a standard value for an 11 kW induction motor. Experimental results of efficiency and power factor are shown in Figures 23 and 24, respectively. As shown, with the air gap length of 0.6 mm, winding #3 consists of the highest efficiency. It somehow consists of lower power factor than the others, with differences increasing from 90% to 92.35% by only changing the stator winding and air gap length. Figures 23 and 24 also show the relative errors comparing the experimental and simulation results. The relative error ε is defined as: where is the experimental value, is the simulation value. It can be seen that the experimental efficiency is close to the simulation results that all the relative errors are less than 2%  Experimental results of efficiency and power factor are shown in Figures 23 and 24, respectively. As shown, with the air gap length of 0.6 mm, winding #3 consists of the highest efficiency. It somehow consists of lower power factor than the others, with differences increasing from 90% to 92.35% by only changing the stator winding and air gap length. Figures 23 and 24 also show the relative errors comparing the experimental and simulation results. The relative error ε is defined as: where is the experimental value, is the simulation value. It can be seen that the experimental efficiency is close to the simulation results that all the relative errors are less than 2% winding #1 winding #2 winding #3 winding #4 Figure 24. Experimental results of power factor versus air gap length for the four stator windings.

Conclusions
The proposed hybrid methodology in this paper is realized by combining FEA and the circuit equivalent circuit model through commercial software MATLAB and ANSYS/MAXWELL to analyze the commercial measures (i.e., changing stator windings and adjusting air gap length) of raising efficiency of induction motors. The methodology is used to compute the starting, overload and full-load performances of induction motors. As an example, it is investigated that a direct-on-line 11 kW DCR induction motor with different air gap lengths and four types of stator windings (i.e., the single-double-layer ∆ winding with 5/6 pitch, the single-layer Y-∆ series winding, the double-layer Y-∆ series winding with an 11/12 pitch, and the double-layer Y-∆ series winding with a 2/3 pitch) thoroughly by the proposed hybrid methodology. The results show that the 11 kW DCR induction motor selecting the double-layer Y-∆ series winding with an 11/12 pitch and 0.6 mm air gap has the highest efficiency. Furthermore, the calculation results are validated by the experimental results of fabricated prototypes. In conclusion, the hybrid methodology presented in this paper is effective to analyze and determine the commercial measures of changing stator winding types and adjusting air gap length to improve the efficiency of induction motors.