Parameter Determination of Asynchronous Machines ... - IEEE Xplore

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 29, NO. 3, SEPTEMBER 2014

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Parameter Determination of Asynchronous Machines From Manufacturer Data Sheet João Marcondes Corrêa Guimarães, Student Member, IEEE, José Vitor Bernardes, Jr., Student Member, IEEE, Antonio Eduardo Hermeto, and Edson da Costa Bortoni, Senior Member, IEEE

Abstract—This paper presents a methodology for obtaining the parameters of the equivalent circuit for an induction motor from the information on the nameplate and manufacturer catalog data sheet. The proposed method is straight and noniterative. The variation in rotor parameters for deep bars or double squirrel-cage rotors is modeled and presented as a function of the slip. The results of remarkable points, such as the starting current, starting and breakdown torques, and rated conditions, are compared with the supplied data, and the comparison showed very good agreement. The methodology was applied to hundreds of motors. A statistical analysis is presented showing that the typical parameters of an equivalent circuit can be readily estimated in the absence of further data. Index Terms—Asynchronous machines, energy efficiency, equivalent circuit, motor parameters.

NOMENCLATURE E Eff gr gx I0 I0p I0q I1 I2 I¯st m1 Mem Mk ¯k M Mst ¯ st M nR nS p P

Magnetizing branch voltage (V). Efficiency of power conversion. Rotor resistance variation factor. Rotor reactance variation factor. No-load current (A). Active no-load current (A). Reactive no-load current (A) Rated stator current (A). Rated rotor current (A). Startup-to-rated current ratio. Number of stator phases. Electromagnetic torque (N·m). Pull-down torque (N·m). Pull-down-to-rated torque ratio. Startup torque (N·m). Startup-to-rated torque ratio. Rated speed (r/min). Synchronous speed (r/min). Number of pole pairs. Mechanical loading power (W).

Manuscript received September 12, 2013; revised November 27, 2013 and March 14, 2014; accepted April 10, 2014. Date of publication May 7, 2014; current version date August 18, 2014. Paper no. TEC-00534-2013. J. M. C. Guimarães, J. V. Bernardes, Jr., and A. E. Hermeto are with the Itajubá Federal University, Itajubá MG 37500-903, Brazil (e-mail: correa. [email protected]; [email protected]; [email protected]). E. C. Bortoni is with Center of Excellence in Energy Efficiency, Institute of Electrical and Energy Systems, Itajubá Federal University, Itajubá MG 37500-903, Brazil (e-mail: [email protected]). Digital Object Identifier 10.1109/TEC.2014.2317525

P2 Pem Pk Prot PF R2 r1 r2 r20 r2k rm s sR sk u V1 x1 x2 x20 x2k xm zr

Rated full-load motor shaft power (W). Electromagnetic power (W). Maximum electromagnetic power (W). Rotational losses (W). Power factor for a given load. Correlation coefficient. Stator resistance (Ω) or (pu). Rotor resistance referred to the stator (Ω) or (pu). Rotor resistance at start-up (Ω) or (pu). Rotor resistance at maximum torque (Ω) or (pu). Magnetizing branch resistance (Ω) or (pu). Motor slip. Rated rotor slip. Rotor slip at the pull-down torque. Coefficient of dependence with the number of poles. Rated phase voltage (V). Stator leakage reactance (Ω). Rotor reactance referred to the stator (Ω) or (pu). Rotor reactance at start-up (Ω) or (pu). Rotor reactance at maximum torque (Ω) or (pu). Magnetizing branch reactance (Ω) or (pu). Reference impedance (Ω). I. INTRODUCTION

HREE-PHASE asynchronous machines, or induction machines, have been widely used since the beginning of the modern industry due to their robustness, easy application, flexibility, ability to work in harsh environments, low cost, and generally better overall performance when the kW-to-r/min ratio is lower than one. Therefore, several methods of calculation, both in steady and dynamic states, have been developed to analyze the machine behavior in relation to its connected mechanical load and its influence on the connected electrical power systems. Some of the concerns in the dynamic domain refer to the evaluation of the start-up time, the voltage drop due to the large start-up currents, and the effect of sequential starting of several machines in an industrial electrical system, among others. In the steady-state domain, it is desirable to model and know the machine behavior under different loading conditions, such as the efficiency in the electromechanical energy conversion, operating power factor, line current, and other electrical and mechanical quantities of interest. Knowledge of performance characteristics is also important when the induction motor works as an asynchronous generator. The equivalent circuit is a suitable model for predicting and evaluating motor performance under such operating conditions [1]–[4]. For an accurate analysis, it is important to consider how

T

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the equivalent circuit parameters vary with frequency for both direct and inverter voltage-fed induction motors. Because the equivalent circuit consists of resistances and inductances, the reactances will depend on the applied voltage frequency, and both the resistance and the reactance may vary with the skin effect in the motor rotor circuit. Both IEEE and IEC standards bring about methods to obtain the parameters of the equivalent circuit for asynchronous machines from tests [5], [6]. Unfortunately, such methods are difficult to implement for large motors and for those already installed in the industry because they demand no-load and lockedrotor testing under a reduced frequency [7]. Online identification techniques [8]–[10] that often use advanced statistics and expert systems [10]–[12] have been proposed to overcome such limitations. However, unless an inverter drive has this calculation embedded in its firmware to achieve better motor control, the analysis of a motor of interest would generally require special instrumentation. In addition, such methods cannot be applied to uninstalled motors, for example, during plant design, in motor selection and specification, and for energy efficiency studies. Except for refurbished machines [13], [14], determining the equivalent circuit parameters for a motor from the nameplate information or from the manufacturer data sheet is a very good approach. Almost all installed motors have a nameplate describing their rated quantities. On the other hand, manufacturers have great interest in supplying information on how their motor operates under different loading conditions: half load, three quarters, and full load. The work of Natarajan and Misra was one of the first reports on determining constant parameters from catalog data [15]. The equivalent circuit parameters were estimated based on information under the above-mentioned loading conditions. At that time, some simplifications were adopted, mainly regarding the calculation of magnetizing branch parameters, neglecting the voltage drop across the stator, saturation, and skin effect. In addition, constant proportionality between stator and rotor leakage reactances was applied according to the NEMA motor design categories. At the same time, Ansuj et al. presented a methodology based on sensitivity analysis [16]. Due to the nonlinear behavior of induction machines, it was difficult to obtain exact and simultaneously practicable equations that relate the performance characteristics and circuit parameters. Therefore, after a rough calculation of the parameters, which were used as initial guesses, slight modifications were applied until a maximum likelihood or minimum variance criterion was reached, thus obtaining the final parameter values. A few years later, Haque presented an iterative procedure based on the solution of several equations derived from the motor equivalent circuit [17]. The magnetizing reactance was calculated from a balance of the reactive power of the motor, stray losses were considered negligible, and frictional losses were assumed to be constant. Again, stator and rotor reactances were divided according to standard motor design categories. The need to consider the rotor parameter variation in doublecage rotors—and with the skin effect in deep-bar squirrel-cage rotors—led some researches to adopt a different model for the

Fig. 1.

Equivalent T circuit of the asynchronous machine.

induction motor, one closer to the circuit commonly applied to synchronous machines and more suitable for application in electromagnetic transient programs [18]–[22]. Modeling of mechanical and stray load losses is discussed in [24], whereas the effects of saturation on the reactances are covered in [18] and [25]–[27]. In general, such methods develop a set of equations that describe the motor behavior as a function of the unknown parameters of the equivalent circuit. An optimization method is applied to iteratively obtain nonnegative parameters to minimize the squared difference between the equation results and the manufacturer data sheet information on some special operating points, such as the rated output power, rated reactive power, breakdown and starting torques, and starting current. The final parameters are obtained when the weighted summation of the absolute errors is smaller than an admissible tolerance. This paper presents a different methodology from the previous ones for obtaining the equivalent circuit parameters of an induction motor from the manufacturer data sheet. The proposed method is straight and noniterative. The rotor parameter variation for deep bars or double cage is modeled and presented as a function of the slip for the conventional equivalent circuit of the motor. The simulation results for remarkable points, such as the starting current, starting and breakdown torques, and rated conditions, were compared with those supplied by the manufacturer, with the comparison showing very good agreement. The methodology was applied to data on hundreds of motors. A statistical analysis was developed, which showed that, in the absence of further data, the typical parameters of an equivalent circuit can be readily obtained as a function of the rated power, voltage, and speed of the motor. II. EQUIVALENT CIRCUIT AND POWER FLOW Fig. 1 shows the equivalent circuit adopted in this paper. Represented in this circuit, known as the T circuit of an induction motor, are the stator parameters, the magnetizing branch parameters, and the rotor parameters referred to the stator. Note that the rotor parameters are variable with the slip to catch the influence of double-cage rotors and the skin effect in deep-bar squirrel-cage rotors. The electromagnetic power and torque can be expressed as a function of the circuit parameters as follows: r

Pem =

m1 V12 s2 2 r r1 + s2 + (x1 + x2 )2

(1)

˜ GUIMARAES et al.: PARAMETER DETERMINATION OF ASYNCHRONOUS MACHINES FROM MANUFACTURER DATA SHEET

691

r

Mem = 2πf1

pm1 V12 s2 . 2 r 2 2 + (x1 + x2 ) r1 + s

(2)

Differentiating the two equations with respect to the slip and making them equal to zero, one can obtain the slip for the maximum torque, which, for a motor, is expressed as r2

sk = r12

+ (x1 +

x2 )2

.

(3)

Fig. 2.

Variation of rotor parameters with the slip.

Fig. 3.

Dynamics of the motor with the proposed parameter variation model.

Therefore, the maximum power and torque, or pull-down torque, will be Pk =

2 r1 +

m1 V12

(4)

r12 + (x1 + x2 )2

and Mk =

pm1 V12 . 2 2 2πf1 r1 + r1 + (x1 + x2 )

(5)

The starting torque, obtained for s = 1, is Mst =

pm1 V12 r2

)2 + (x + x )2 2πf1 (r1 + r20 1 20

.

(6)

This equivalent circuit also represents the power flow in the asynchronous machine. The power dissipated in r1 represents the copper loss in the stator coils. The power dissipated in rm represents the losses due to eddy currents and hysteresis in the stator core. Sometimes, rm also includes the windage, friction, and stray losses, which are all considered here. As long as r2 /s can be divided into r2 and r2 (1−s)/s parcels, the power dissipated in r2 represents the rotor copper loss and the power dissipated in r2 (1−s)/s is, eventually, the mechanical power available in the motor shaft. The electromagnetic power is the input power minus the summation of the stator copper loss with the so-called rotational losses, which include hysteresis, eddy currents, windage, friction, and stray losses. The electromagnetic power crosses the machine air gap into the rotor and is divided into two parcels: a smaller one proportional to the rotor slip(s), the rotor copper loss, and a major one, which is its complement, proportional to (1−s), which is the mechanical power available in the motor shaft. The equivalent circuit presented, which has constant parameters and is more suitable for wound rotor induction motors, has the disadvantage of providing a high operational slip, with higher rotor losses, when a high starting torque is required and vice versa. To obtain a high starting torque with low slip operation, double-cage or deep-bar squirrel-cage rotors provide variable rotor parameters due to the skin effect, as depicted in the Appendix, roughly represented in Fig. 2. The theory states that the rotor parameters vary as a function of the square root of the slip [3], [4]. Therefore, a monotonic nonlinear function of the slip should be defined to model this variation. Examples of possible functions are described in (7)

to (10) [22], [23]. The most adequate function can be chosen to meet the pull-up torque requirements √

r2 (s) = r20 exp gr 1 − s (7) √

x2 (s) = x20 exp gx 1 − s (8) and

r2 (s) = r20 0.5 + 0.5 s/sk x2 (s) = x20 0.4 + 0.6 sk /s .

(9) (10)

In this paper, (7) and (8) are adopted to represent the rotor parameter variation due to the skin effect, rather than using high-order models with two or more rotor branches, which are more suitable for use with electromagnetic transient programs [18]–[22]. Fig. 3 shows the flexibility of the proposed parameter variation model. In Fig. 3 (a), the typical torque curve shape is obtained from four simulations considering different ratios of start-up torque to rated torque (100%, 150%, 200%, and 250%), representing four starting torque motor categories. In all cases, the ratio of the pull-down torque to the rated torque is 200%.

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Fig. 3 (b) shows four simulations for starting-to-rated-current ratios of 400%, 500%, 600%, and 700%. III. PARAMETER DETERMINATION The aim of this paper is to present a novel methodology for obtaining the equivalent circuit parameters of a machine from its nameplate and manufacturer data sheet. The nameplate of a motor often contains information about some rated and fullload quantities, such as the mechanical shaft power, voltage, current, speed, efficiency, power factor, and starting current. The manufacturer data sheet, on the other hand, provides additional information related not only to the rated condition but also to other conditions, such as the efficiency and power factor at half load and at three quarters. The relations between the startingto-rated current, the starting-to-rated torque, and the pull-downto-rated torque are also supplied. The following items show how the parameters are determined. Methods for determining the parameters based only on the nameplate information are presented. The value of additional information from the manufacturer data sheet is considered in the improved formulation.

In every operating condition, there is a power balance in the motor in which the electromagnetic power is the result of subtracting the rotational losses and the stator copper loss from the power input P2 P2 − . Ef f 1−s

3r2 I12 + K = P2

(11)

The rotational loss is the summation of the core losses (hysteresis and eddy currents), windage, friction, and stray losses. Despite the shaft speed changes with the machine loading, the summation of the aforementioned losses is practically constant for all the loading conditions, mainly due to the constant rotation speed of the revolving field [4]. In other words, as long as the windage and friction losses decrease with the increasing loading (shaft speed reduction), the core and stray losses increase in almost the same proportion, such that the rotational losses are kept nearly constant for all the machine loadings in the operating region. Therefore, (11) can be written for the three operating conditions of the motor, i.e., half of the full load, three quarters of the full load, and full load. The stator resistance is the slope of the linear fit obtained from ordinary least-squares techniques

P2 P2 2 − ; 3I1 . (12) r1 = slope Ef f 1−s The slip for any loading condition between no load and full load can be obtained as 1 (13) 1 − 1 − 4sR (1 − sR ) P/P2 . s= 2 For the rotor resistance, it is considered that the power dissipated in the resistance due to the current I2 is lower than the power dissipated in the resistance due to the current I1 by a

s . 1−s

(14)

Again, writing this balance for the three known operating conditions of the motor, the rotor resistance referred to the stator can be obtained as the slope of the equation by using leastsquares linear fit techniques

s ; 3I12 . r2 = slope P2 (15) 1−s The value of the rotor resistance referred to the stator at the start-up can be determined by considering that all the electromagnetic power in this condition is dissipated in the rotor resistance ¯ st nS P2 M = . (16) r20 2 3 I2 I¯st nR If only nameplate information is available, the stator resistance per phase may be determined by handling (3) and (5), yielding r1 =

A. Determination of r1 , r2 , and r20

3r1 I12 + Prot =

constant amount K,

45V12 r − 2. πMk ns sk

(17)

The rotor resistance referred to the stator is determined by knowing that its variable parcel dissipates rated power at rated conditions sR P2 2 I . r2 = (18) 1 − sR 3 2 The rated rotor current is calculated based on the full-load power factor (19) I2 = I1 P F 1 + (sR /sk )2 with sR and sk given by ns − nR nS 2 ¯ ¯ sk = sR Mst + Mst − 1 .

sR =

(20) (21)

From (7), the rotor resistance variation factor is ln (r /r ) gr = √ 20 2 . 1 − sR

(22)

B. Determination of x1 , x2 , and x20 Analyzing (3), it is possible to disregard the squared reactive term, resulting in a simplified expression to determine x2k when s is equal to sk , x2k = with

r2k sk

√ r2k = r20 exp gr 1 − sk .

(23)

(24)

By using (5) and (6), it is possible to derive an expression that relates x20 to x2k as a function of the maximum and start-up


torques, as follows: x20 =

x M ¯ k /M ¯ st − r2 . 2r2p 2p 2k

(25)

With x2p and x2k on hand, one can calculate the rotor reactance variation factor and the reactance x2 for the rated slip gx = and

ln (x20 /x2k ) √ 1 − sk

√ x2 = x20 exp gx 1 − sR .

(26)

(27)

The parameter x1 is obtained to match the starting current 2 V1 x20 )2 − x1 = − (r + r . (28) 1 20 I1 I¯st 1 + (sR /sk )2 C. Determination of rm and xm The rotational losses can be calculated from (11) for the full load as P2 P2 − − 3r1 I12 . Pr ot = (29) Ef f 1−s The magnetizing resistance is determined by considering that it will assume all the rotational losses rm = 3E 2 /Pr ot .

(30)

where E is obtained from the voltage drop in the stator impedance (31) E = V1 − I1 r12 + x21 . The no-load current is considered as the reactive component of the rated stator current. Therefore, its active and reactive components are I0p = E/rm 2 . I0q = I12 sin2 [acos (F P )] − I0p

In addition, the use of a spreadsheet brings many advantages, such as ease of programming and software prototyping, easy data handling and sorting, and graph presentation capabilities. On the other hand, it is difficult to deal with complex numbers without working with macros; thus, dynamic simulation was done by using MATLAB resources. Of course, the presentation of the 2980 results in tabular form is neither intelligible nor productive. Therefore, the quality of the parameters obtained with the application of the proposed methodology is presented as the accuracy of the calculation of some remarkable points in relation to the supplied information. The starting current; starting, breakdown, and rated torques; and slip for the breakdown torque were calculated and compared with the supplied data. The results of this analysis are depicted in the graphs in Figs. 4–8, which show the supplied data in the abscissas and the calculated data in the ordinates. A linear fit with origin forced intercept is applied to obtain the correlation between the two datasets. The slope of the straight line shows the average ratio between the calculated quantities in relation to its supplied value. The closer the slope is to 45◦ , the more precise the calculated values are compared with the supplied ones. The correlation coefficient (R2 ) is a global measure of the deviation of each value from the calculated straight line. The closer this value is to the unit, the closer the calculated values agree with the supplied ones. A very good agreement can be observed between the calculated values in comparison with the supplied information. Although there is minor divergence in the rated and pull-down results, the calculated starting quantities are exact. This happens because after the resistances have been calculated by (12), (15), and (16) to meet the starting torque and efficiencies during regular operation, the reactances are calculated by (25), (27), and (28) to meet the pull-down torque and starting current.

(32)

V. EVALUATION OF TYPICAL PARAMETERS

(33)

The determination of the typical parameters of induction motors must be carried out with extreme care. It is well known that different manufacturers adopt different design approaches to obtain the specified remarkable points in desired efficiency and power factor levels. Even for the same manufacturer, high-efficiency motors have a different design from that of standard-efficiency units. For the same motor power, different starting torque categories are obtained by using different design criteria. Therefore, one cannot obtain a typical value for the stator resistance if both standard- and high-efficiency motors are enrolled in the same universe under analysis. The same rule applies for the starting rotor resistance; a typical value could not be obtained if motors of different starting categories are being compared. Of course, average values for equivalent circuit parameters could always be obtained from a comparison of motors for different purposes. Nevertheless, such values will not have any physical meaning because they will represent neither a standard- nor a high-efficiency motor, or neither a high- nor a regular-starting-torque motor category.

The magnetizing reactance is given by xm = E/I0q .

693

(34)

IV. EVALUATION OF THE OBTAINED PARAMETERS The presented model was applied to a universe of 2980 different motor data obtained from online and PDF catalogs of manufacturers. The data covered both IEC and NEMA standards, rated voltages from 220 to 575 V, 50 and 60 Hz, 1 to 4 pole pairs, and rated power ranging from 0.12 to 675 kW. The methodology was implemented by using regular spreadsheet software. For a single motor, all the input data, calculations, and results occupy a single line, spanning almost 70 columns. The adoption of this procedure made possible the insertion of data on many motors, one for each line of the spreadsheet. Therefore, the results can be obtained with a simple double click at the end of the line containing the cells of the calculations; all the calculations are repeated for all the motors.

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Fig. 4.

Fig. 5.


Calculated rated torque (M R ∗).

Calculated pull-down torque (M k ∗).

Fig. 8.

Calculated slip for pull-down torque (sk ∗).

The development of rules to describe the variation in equivalent circuit parameters as a function of the motor power is of great interest not only in readily estimating parameter values but also in comparing design criteria between manufacturers. Therefore, a study was carried out to analyze the nature of the variation in the parameters obtained from the application of the described equations, as a function of the motor output. In power systems and in studies of electrical machines, it is common to use a base impedance, which is a function of the rated voltage and the rated electrical power, to obtain per unit values. Considering the aim of presenting the per-unit parameter variation as a function of the motor power, the adoption of a base impedance function of the power would be redundant. However, the studies have shown a clear dependency of the stator reactance and of the magnetizing branch parameters on the number of pole pairs. Therefore, to obtain a single expression for each parameter, the reference impedance is defined to take into account this influence zr =

Fig. 6.

Calculated starting torque (M P ∗).

Fig. 7.

Calculated starting current (IP ∗).

V12 . (2p)u 104

(35)

A least squares algorithm was applied to identify the values of u, yielding –0.25 for rm , 0.333 for xm , and −0.333 for x1 . These values must be kept constant for all the analyzed motors. Hence, the typical per-unit parameters of standard efficiency, 60 Hz, and NEMA motor from a given manufacturer would result in the models presented in Figs. 9–12. The high correlation coefficients obtained in the regressions show the accuracy of the presented model and of the adopted approaches. Once the per-unit parameters have been obtained from the presented relationships, their values in ohms can be readily calculated by applying the reference impedance described in (35). The presented model may also be used either for comparing design approaches between different manufacturers or for comparing design criteria between standard and energy-efficient motors from the same manufacturer. Fig. 13 presents a comparison of the per-unit resistances of two 50-Hz NEMA motors, with two pole pairs, from the same manufacturer. The solid line in the images represents a standard-efficiency motor design, and the dashed line represents a high-efficiency motor design. As shown in Fig. 13 (a), the stator resistance of the high-efficiency motor is about 80% lower than that of the standard-efficiency motor. This relationship is about 70% for


Fig. 9.

Stator per unit resistance (a) and reactance (b).

Fig. 10.

Rotor per unit resistance (a) and reactance (b) referred to stator.

Fig. 11.

Starting rotor per unit resistance (a) and reactance (b) referred to stator

Fig. 12.

Magnetizing branch per unit resistance (a) and reactance (b).

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Fig. 14.

Skin effect in deep bars placed in the rotor iron core.

VI. CONCLUSION This paper presented a novel methodology for obtaining the parameters of the equivalent circuit of induction motors from its nameplate and the manufacturer data sheet. The presented methodology is straight and noniterative. The adoption of rotor parameters that are variable with the slip made it possible to include the skin and saturation effects to accurately reproduce both the starting, peak, and rated quantities by using a conventional equivalent circuit. Obtaining the equivalent circuit parameters from a given motor data sheet is very interesting as long as its electrical, mechanical, and energy behavior can be studied even before it is purchased or installed in a system. With the adoption of the proper reference impedance, it was possible to obtain the per-unit values of several parameters with predictable behaviors as a function of the motor power. Therefore, the typical parameters could be obtained from simple relationships, allowing for average motor simulation. Given the observed characteristics, the developed model could also be used to evaluate the motor design criteria used by a single manufacturer or to compare the criteria used between several manufacturers. APPENDIX SKIN EFFECT AND SATURATION

Fig. 13. motors.

Stator and rotor per unit resistance of standard and energy-efficient

the rotor resistance, as presented in Fig. 13 (b). The reduction in resistances provides a lot of information on the approach adopted by this manufacturer to obtain a high-efficiency motor, that is, by using more copper on the wirings, which allows for reduced stator losses both in the stator and in the rotor. Fig. 13 (c) shows the behavior of the magnetizing branch resistance, which is decreased by about 45%. In general, this reduction involves the use of thinner lamination and better magnetic material in the machine core construction. In all the cases, the difference is more significant for motors with lower power than for those with higher power, as expected.

Skin effect and saturation are phenomena that must be incorporated into motor models to correctly represent the dynamic processes imposed on the machine. The skin effect is particularly important in the double-cage and the deep-bar squirrel-cage rotors of induction motors. Fig. 14 shows the deep bar placed in the rotor iron core in two instances: during start-up (a) and in normal operation (b). When the rotor is locked, the frequency of its induced voltage is equal to the stator voltage frequency. The reluctance of the air gap at the top of the rotor bar is greater than the reluctance of the rotor iron core. As long as the inductance is inversely proportional to the reluctance and the reactance is the product of the inductance and the angular frequency, the reactance at the top of the bar will be smaller than the reactance inside it. Therefore, the current density at the top of the rotor bar will be greater than that inside it (a). This current density in a reduced cross-section area will result in a high rotor resistance value and a high starting torque. During normal operation with lower slips, the frequency of the rotor-induced voltage tends toward zero and the reactance can be neglected. Better current distribution throughout the cross


section of the bar will be observed, resulting in smaller rotor resistance (b). The rotor magnetic field will be subject to the low reluctance of the iron core, leading to higher inductance. Therefore, whereas the rotor resistance is proportional to the slip, the inductance will increase with the slip reduction. The variation from state (a) to state (b) is smooth and continuous. The theory holds that the rotor parameters vary as a function of the square root of the slip [3], [4]. On the other hand, the high start-up current in the stator and rotor windings can result in saturation, further reducing the magnetizing inductance and leakage inductances. Studies have been carried out to understand in depth the saturation phenomenon in induction motors [18], [25]–[27]. Although saturation was not the subject of this paper, the and x20 ) of the equivalent circuit starting rotor parameters (r20 of a given motor were calculated to meet the starting torque and starting current. In the same way, the rated rotor parameters (r2 and x2 ) were calculated to meet the rated conditions. Therefore, during motor simulations, the rotor parameters should vary from their starting values to their rated values, replicating the actual motor operation such that the skin and saturation effects are included in the chosen law of parameter variation with the slip. ACKNOWLEDGMENT The authors would like to thank FAPEMIG, CNPq, CAPES, FINEP, and INERGE for their support in the conduct of this research, and to Dr. E. Jacobson for his contributions. REFERENCES [1] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Machinery. New York, NY, USA: IEEE Press, 1995. [2] A. E. Fitzgerald, C. Kingsley, Jr., and S. D. Umans, Electric Machinery. New York, NY, USA: McGraw-Hill, 1992. [3] M. Kostenko and L. Piotrovsky, Electrical Machines—Part 2. Moscow, USSR: Mir Pub., 1977. [4] P. L. Alger, Induction Machines, Their Behavior and Uses, 2nd ed. New York, NY, USA: Gordon and Breach, 1965. [5] Test Procedure For Polyphase Induction Motors and Generators. IEEE Standard 112, 2004. [6] Rotating Electrical Machines. Part 2-1: Standard Methods for Determining Losses and Efficiency From Tests, IEC Standard 60034-2-1, 2007. [7] E. C. Bortoni, “Are my motors oversized?,” Energy Convers. Manag., vol. 50, no. 9, pp. 2282–2287, Sep. 2009. [8] M. S. Zaky, M. M. Khater, S. S. Shokralla, and H. A. Yasin, “Wide-speedrange estimation with online parameter identification schemes of sensorless induction motor drives,” IEEE Trans. IE, vol. 56, no. 5, pp. 1699–1707, 2009. [9] M. Wlas, Z. Krzemin ski, and H. A. Toliyat, “Neural-network-based parameter estimations of induction motors,” IEEE Trans. IE, vol. 55, no. 4, pp. 1783–1794, 2008. [10] C. Grantham and D. J. McKinnon, “Rapid parameter determination for induction motor analysis and control,” IEEE Trans. Ind. Appl., vol. 39, no. 4, pp. 1014–1020, Jul./Aug. 2003. [11] P. Pillay, R. Nolan, and T. Haque, “Application of genetic algorithms to rotor parameter determination from transient torque calculations,” IEEE Trans. Ind. Appl., vol. 33, no. 5, pp. 1273–1282, Sep./Oct. 1997. [12] V. P. Sakthivel, R. Bhuvaneswari, and S. Subramanian, “Multi-objective parameter estimation of induction motor using particle swarm optimization,” Eng. Appl. Artif. Intell., vol. 23, pp. 302–312, 2010. [13] E. C. Bortoni, J. Haddad, A. H. M. Santos, E. M. Azeredo, and R. A. Yamachita, “Analysis of repairs on three-phase squirrel-cage induction motors performance,” IEEE Trans. Energy Convers., vol. 22, no. 2, pp. 383–388, Jul. 2007.

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[14] A. G. Siraki, P. Pillay, and P. Angers, “Full load efficiency estimation of refurbished induction machines from no-load testing,” IEEE Trans. Energy Convers., vol. 28, no. 2, pp. 317–326, Jun. 2013. [15] R. Natarajan and V. K. Misra, “Parameter estimation of induction motors using a spreadsheet program on a personal computer,” Elect. Power Syst. Res., vol. 16, pp. 157–164, 1989. [16] S. Ansuj, F. Shokooh, and R. Schinzinger, “Parameter estimation for induction machines based on sensitivity analysis,” IEEE Trans. Ind. Appl., vol. 25, no. 6, pp. 1035–1040, Nov./Dec. 1989. [17] M. H. Haque, “Estimation of three-phase induction motor parameters,” Elect. Power Syst. Res., vol. 26, pp. 187–193, 1993. [18] D. Lindenmeyer, H. W. Dommel, A. Moshref, and P. Kundur, “An induction motor parameter estimation method,” Elect. Power Energy Syst., vol. 23, pp. 251–262, 2001. [19] J. Pedra, “Estimation of induction motor double-cage model parameters from manufacturer data,” IEEE Trans. Energy Convers., vol. 19, no. 2, pp. 310–317, Jun. 2004. [20] J. Pedra, “Estimation of typical squirrel-cage induction motor parameters for dynamic performance simulation,” in Proc. IEEE Generation, Transmiss. Distrib., vol. 153, no. 2, pp. 137–146, Mar. 2006. [21] J. Pedra, “On the determination of induction motor parameters from manufacturer data for electromagnetic transient programs,” IEEE Trans. Power Systems, vol. 23, no. 4, pp. 1709–1718, Nov. 2008. [22] M. H. Haque, “Determination of NEMA design induction motor parameters from manufacturer data,” IEEE Trans. Energy Convers., vol. 23, no. 4, pp. 997–1004, Dec. 2008. [23] J. S. de Sá, “Contribution to the analysis of the thermal behavior of threephase squirrel cage induction motors,” Ph.D. dissertation, Universidade Estadual de Campinas, Campinas, São Paulo, Brasil, 1989. [24] M. Torrent, “Estimation of equivalent circuits for induction motors in steady state including mechanical and stray load losses,” Eur. Trans. Elect. Power, vol. 22, pp. 989–1015, 2012. [25] J. Pedra, I. Candela, and A. Barrera, “Saturation model for squirrel-cage induction motors,” Elect. Power Syst. Res., vol. 79, pp. 1054–1061, 2009. [26] G. J. Rogers and D. Shirmohammadj, “Induction machine modelling for electromagnetic transient program,” IEEE Trans. Energy Convers., vol. 2, no. 4, pp. 622–628, 1987. [27] L. Monjo, F. Córcoles, and J. Pedra, “Saturation effects on torque- and current–slip curves of squirrel-cage induction motors,” IEEE Trans. Energy Convers., vol. 28, no. 1, pp. 243–254, Mar. 2013. João Marcondes Corrêa Guimarães (S’12) was born in Santos, Brazil, on January 29, 1990. He is currently working toward the undergraduate degree in power systems engineering from Federal University of Itajubá (UNIFEI), Itajubá, Brazil. His research interests include electrical machine design, energy efficiency, and applied sensors. He is a Member of the Tutorial Educational Program, Electrical Engineering Institute, where he develops several researches related to efficiency measurement in synchronous and asynchronous machines. José Vitor Bernardes, Jr. (S’13) was born in Carmo de Minas, Brazil, on December 30, 1988. He received the graduate degree from UNIFEI, Itajubá, Brazil, as a Power Systems Engineer in 2012 and is working toward the Master’s degree in electrical machines. His research interests include power systems dynamics, electrical machine design, and parameter identification. Antonio Eduardo Hermeto was born in Lavras, Brazil, on April 11, 1941. He received the graduate degree from the Electromechanical Institute of Itajubá, IEI (now UNIFEI), Itajubá, Brazil. He became an Electromechanical Engineer in 1966. He is currently the Former Director of the Electrical Engineering Institute of UNIFEI. His research interests include power systems analysis, electrical machines, and systems dynamics. Edson da Costa Bortoni (S’94–M’96–SM’05) was born in Maringá, Brazil, on December 1, 1966. He received the electrical engineering degree from UNIFEI, Itajubá, Brazil, in 1990; the M.Sc. degree in energy systems planning from the University of Campinas, Campinas, Brazil, in 1993; the D.Sc. degree in power systems from the University of São Paulo, São Paulo, Brazil, in 1998; and the L.D. degree from USP, São Carlos, Brazil, in 2012. He is currently a Professor at UNIFEI. His research interests include instrumentation, power generation, and energy systems. He is a Fellow of the ISA.