THREE-PHASE induction motors are widely used in industries

Transcription

1 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 23, NO. 4, DECEMBER Determination of NEMA Design Induction Motor Parameters From Manufacturer Data M. H. Haque, Senior Member, IEEE Abstract This paper proposes a simple method of determining the equivalent circuit parameters of National Electrical Manufacturers Association (NEMA) design A and B types of induction motors from standard manufacturer data such as rated output power, starting torque, breakdown torque, and efficiency and power factor at rated output power. A set of nonlinear equations for various quantities is first derived from the equivalent circuit with a singlecage rotor model, and then, equate to the corresponding actual values supplied by the manufacturer. These equations are then solved using a least-squares based algorithm to determine the motor parameters. The rotor parameters are considered as slip dependent to predict the starting torque of the motor and that requires refining the breakdown torque equation as well as the slip at which the breakdown torque occurs. The proposed method of determining the motor parameters is then tested on more than 300 large-size HV induction motors. The effectiveness of the proposed method is evaluated by calculating various external quantities of the motors through the estimated parameters and comparing them with the corresponding actual values supplied by the manufacturer. Index Terms Equivalent circuit, induction motor, manufacturer data, National Electrical Manufacturers Association (NEMA) design, parameter estimation. I. INTRODUCTION THREE-PHASE induction motors are widely used in industries because of their lower price, ruggedness, and almost maintenance-free operation. In fact, more than 50% of total electrical energy generated in the United States is consumed by integral horsepower (hp) three-phase induction motors [1]. The performance characteristics of an induction motor are usually determined from its equivalent circuit. Unfortunately, the manufacturers do not provide the equivalent circuit parameters instead supply some catalog data such as rated output power, starting torque and current, breakdown torque, and other quantities (current, power factor, efficiency, torque, and speed) at rated output power. The most commonly used equivalent circuit of an induction motor (with a single-cage rotor model) has six fixed parameters [2] [4]. Such an equivalent circuit can predict the motor characteristics from normal running speed down to breakdown speed, but it provides erroneous results at starting point and during accelerating period (or at lower speeds) [5] [10]. However, the torque slip (T s) characteristic at lower speeds plays an important role in the analysis of stalling and/or reacceleration process of a loaded motor following a fault or during voltage Manuscript received April 11, 2007; revised January 27, Current version published November 21, Paper no. TEC The author is with the Center for Smart Energy Systems, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore , Singapore ( emhhaque@ntu.edu.sg). Digital Object Identifier /TEC sag condition [11]. In order to get better results at starting or lower speeds, it is necessary to consider the equivalent circuit with a double-cage rotor model [5] [10]. In some cases, a single set of parameters that can predict the torque slip, current slip, and power factor slip characteristics over the full range of slips (even with a double-cage rotor model) may not exist [8]. The selection of equivalent circuit may depend on study objective. For example, when the starting characteristics are of interest, it is necessary to use the double-cage rotor model. On the other hand, when the motor characteristics from synchronous speed down to breakdown speed are of interest, a single-cage rotor model is usually sufficient. The parameters of the equivalent circuit of an induction motor (with a single-cage rotor model) are usually determined from some test data such as stator resistance measurement, no-load, and locked-rotor test data [2], [3]. The similar test data cannot be used to determine the parameters of a double-cage rotor model. The motor parameters can also be determined from manufacturer data to satisfy some external characteristics. Several methods of determining the motor parameters, for both singleand double-cage rotor models, from manufacturer data are reported in the literature [5], [6], [8], [10], [12], [13]. In the normal operating region, the frequency of rotor current is very low and the IEEE Standards 112 [2] recommends to carryout the locked-rotor test at a lower frequency. The rotor parameters determined at a lower frequency are not significantly different from dc values and can faithfully be used in the slip range 0 s max, where s max is slip at the maximum or breakdown torque. However, when the rotor frequency increases at higher slip, the combination of skin effect and cross-slot flux significantly change the rotor parameters, especially for a motor with large cross-section rotor bars [14]. References [15] [17] also observed that the rotor parameters are not constant but slip dependent. Use of fixed rotor parameters in the conventional equivalent circuit with a single-cage rotor model is the main reasons of providing erroneous results at starting or higher slips. Reference [14] intelligently incorporated the effects of rotor frequency on rotor parameters by adjusting their values with slip through empirical formulas to predict the starting torque of National Electrical Manufacturers Association (NEMA) design B motors. With the aforementioned adjustment, a single-cage rotor model can provide similar torque slip characteristic to that of a double-cage rotor model. However, the empirical adjustment of rotor parameters, as suggested in [14], has not sufficiently been investigated and reported in the literature, and that motivates the author to conduct this study. This paper proposes a simple method of determining the equivalent circuit parameters of an induction motor, with /$ IEEE

2 998 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 23, NO. 4, DECEMBER 2008 Fig. 1. Equivalent circuit of an induction motor. (a) Single-cage rotor model. (b) Double-cage rotor model. slip-dependent rotor parameters, from the manufacturer data. The proposed method is tested on more than 300 HV motors of various sizes. The parameters found by the proposed method are then used to predict various external quantities of the motors and compare them with the corresponding actual values supplied by the manufacturer. II. INDUCTION MOTOR MODEL The single-phase equivalent circuit of a three-phase induction motor, with a single-cage rotor model, is shown in Fig. 1(a), where R 1,X 1,R 2,X 2,R c, and X m represent the stator resistance, stator leakage reactance, rotor resistance, rotor leakage reactance, core loss resistance, and magnetizing reactance, respectively. Note that the aforementioned circuit can predict the motor characteristics in the normal operating region, but provides erroneous results at starting [5], [6], [14], [15]. In order to get better starting characteristics, a double-cage rotor model is to be used. The commonly used equivalent circuit for such a model is shown in Fig. 1(b), where the parameters of the inner (outer) cage are represented by a subscript i ( o ). In the aforementioned circuits, the rotor quantities are transferred to the stator. As mentioned earlier, Fig. 1(a) can provide similar torque slip characteristic to that of Fig. 1(b) when the rotor parameters (R 2 and X 2 ) of Fig. 1(a) are carefully adjusted with slip. Such an equivalent circuit of a single-cage rotor model with slipdependent rotor parameters is separately shown in Fig. 2(a) and is used in this study. The empirical adjustment of rotor parameters of Fig. 2(a), for NEMA design B motors, is described in [14]. The same adjustment is also used in this study and is given in the following. When s>s max ( ) s R 2 (s) = R 20 = fr (s)r 20 (1) X 2 (s) = ( s max smax s ) X 20 = fx (s)x 20 (2) Fig. 2. Single-cage rotor model with slip-dependent rotor parameters. (a) General equivalent circuit. (b) Thevenin equivalent circuit. TABLE I MINIMUM ALLOWABLE TORQUE RATIOS OF NEMA DESIGN A AND BMOTORS when s s max R 2 (s) =R 20 and X 2 (s) =X 20. (3) Here, R 20 and X 20 are the rotor resistance and leakage reactance at low frequency and s max is slip at the maximum or breakdown torque. Note that motors of different NEMA design classifications may need different empirical adjustments of rotor parameters, but that aspect has not been investigated in this study. The Thevenin equivalent of Fig. 2(a) is shown in Fig. 2(b) where the Thevenin parameters (V th,r th, and X th )aregiven by V th = V 1 Z sh Z 1 + Z and Z th = Z 1Z sh =(Rth + jx th ). sh Z 1 + Z sh (4) Here, Z 1 =(R 1 + jx 1 ) and Z sh = R c // jx m.thevalueof s max used in (1) and (2) is given by s max = R 20 R 2 th +(X th + X 20 ) 2. (5) NEMA design classifications are based on the minimum allowable ratios of starting toque to full-load torque (T st /T FL ) and breakdown (or maximum) torque to full-load torque (T max /T FL ). The ratios are defined in NEMA Standards MG-1 [18]. Note that the ratios for NEMA design B motors are not separately specified but combined with design A motors. Typical values of torque ratios for NEMA design A and B motors of various sizes are listed in Table I. The last column of the table is obtained from the second and third columns. It can be observed in the table that the ratio of minimum starting

3 HAQUE: DETERMINATION OF NEMA DESIGN INDUCTION MOTOR PARAMETERS FROM MANUFACTURER DATA 999 torque to minimum breakdown torque (T st /T max ) for large size motors ( 100 hp) is less than or equal to III. DETERMINATION OF EQUIVALENT CIRCUIT PARAMETERS Most of the previous studies of determining the motor parameters from manufacturer data ignored the core loss. Results of such studies may not be used to predict the motor efficiency [5], and hence, current and power factor. However, estimation of motor efficiency is very important in many applications such as energy management [19], [20]. An induction motor has four components of losses: core loss (P core ), friction and windage loss (P fw ), stray-load loss (P stray ), and copper loss (P cu ) in the stator and rotor windings. In general, P core and P fw are considered as more or less constant in the normal operating region, but P stray and P cu depend on load. In most cases, P stray is neglected because of its relative magnitude and variable nature. The sum of P core and P fw in the normal operating region is considered as constant loss (P const ). In this study, P const is first determined at full-load condition, and then, split into P core and P fw using a predefined constant. It has been reported that the equivalent circuit of an induction motor, with a single-cage rotor model, has four independent parameters when the core loss is ignored [9]. However, when the core loss is considered, it should have five independent parameters. With a fixed relationship between P const and P core, it can be considered that the circuit has four independent parameters, but that requires determination of another independent variable P const. Thus, determination of equivalent circuit parameters of Fig. 2(a) involves finding the values of seven variables (six parameters and P const ) out of which only five are independent. This requires imposing two relationships or constraints between the variables. Otherwise, the problem is not well defined. It has been reported that the motor characteristics are least sensitive to the stator resistance (R 1 ), and thus, R 1 can be considered as [5] R 1 = α r R 20. (6) Here α r is a positive constant. As mentioned, P const is determined at full-load condition, and then, split into P core and P fw. Thus, the second constraint can be considered as P core = α c P const P fw =(1 α c )P const. (7) Here α c (<1) is a positive constant. By knowing P core,the value of R c can be evaluated as R c = 3E2 (s FL ). (8) P core Here, E(s FL ) is the voltage across the shunt branch of Fig. 2(a) at full-load slip (s FL ). With constraints (6) and (7), the problem ultimately transformed into the determination of four motor parameters (R 20,X 1,X 20, and X m ) and the constant loss P const.inthis study, the aforementioned five variables are determined from the following five independent pieces of information that can easily be obtained from the manufacturer s catalog: 1) mechanical output power at full load (Po FL ); 2) maximum or breakdown torque (T max ); 3) starting torque (T st ); 4) input active power at full load (Pin FL ); and 5) input reactive power at full load (Q FL in ). Note that the first three information are directly specified in the catalog and the last two information can easily be extracted from the full-load efficiency (η FL ) and power factor (cos ϕ FL ) that are also specified in the catalog P FL in = P o FL and Q FL in = Pin FL tan ϕ FL. (9) η FL At full load, the stator line current (I FL ) and electromagnetic developed torque (T FL ) can be expressed as I FL = P FL o 3VηFL cos ϕ FL and T FL = 60(P FL o + P fw ) 2πn FL. (10) Here, V is the rated line voltage and n FL is the motor speed [in revolutions per minute (r/min)] at full load. Thus, the five independent variables (X 1,R 20,X 20,X m, and P const ) can easily be determined from five independent pieces of information (Po FL, Pin FL the solution of the following nonlinear equations, QFL in, T max, and T st ) and that requires f 1 (x) =Po FL P o (s FL )=0 (11.1) f 2 (x) =Pin FL P in (s FL )=0 (11.2) f 3 (x) =Q FL in Q in (s FL )=0 (11.3) f 4 (x) =T max T (s max )=0 (11.4) f 5 (x) =T st T (1) = 0. (11.5) The previous equations can be rewritten in a general form as F(x) =0. (12) Here F =(f 1,f 2,f 3,f 4,f 5 ) T and x =(X 1,R 20,X 20,X m, P const ) T. The expressions for P o (s FL ), P in (s FL ), Q in (s FL ), T (s max ), and T (1) are given in the Appendix. Note that T (s max ) and T (1) are the internal torques (or the electromagnetic developed torques) that include P fw. Thus, values of T max and T st specified by the manufacturer are also to be adjusted accordingly. The set of nonlinear equations (12) can be solved using the fsolve routine given in the Optimization Toolbox of MATLAB [21]. It uses nonlinear least-squares algorithm that employs the Gauss Newton or the Levenberg Marquardt method. The advantage of a least-squares-based method is that when the system of equations does not have a zero, the method converges to a point where the residual is small. The residual (ε) at the converged point can be considered as ε = fi 2. (13) The selection of initial values of unknown is very important in all iterative calculations. In this study, the following initial values are used and are found to be satisfactory for all the i

4 1000 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 23, NO. 4, DECEMBER 2008 cases studied R 20 = s FLPo FL 3I1FL 2 X 1 = X 20 = 0.05V 1 I 1FL TABLE II RMS ERROR IN VARIOUS QUANTITIES X m = V 1 P const =0.03Po FL. 0.2I 1FL Here, V 1 is the rated phase voltage and I 1FL is the stator phase current at full load. The effectiveness of the proposed method of determining the motor parameters can be evaluated by calculating various external quantities of the motor through its equivalent circuit and comparing them with the corresponding actual values supplied by the manufacturer. The error in the calculated values can considered as calculated value actual value error =. (14) actual value The rms error for n number of case studies can be defined as n i=1 rms error = error2. (15) n Note that the manufacturer may provide more information or data in the catalog. However, the data that are not used in formulating the problem may not be predicted correctly. Thus, the problem formulation or determination of motor parameters may depend on the study objective. In this study, the torque slip characteristic is given higher priority because it is widely used in analyzing the stalling and reacceleration process of a loaded motor following a disturbance or during voltage sag condition as well as in voltage stability analysis of a power system [11]. IV. PRELIMINARY RESULTS The proposed method of determining the motor parameters from manufacturer data is vigorously tested on a large number of HV induction motors. The catalog data of the motors are obtained from the manufacturer Web site [22]. The rated voltage and output power of the motors are kv and hp (or kw), respectively. According to Table I, the ratio of minimum starting torque to minimum breakdown torque for large size motors ( 100 hp) is less than or equal to In this study, only the motors that satisfy the aforementioned criterion are considered and that covers most of the NEMA design A and B types of motors. More than 325 motors are found in this category and the proposed method is applied to all of them in determining the parameters. The values of α r and α c in (6) and (7) are considered as 1.5 and 0.5, respectively. The method converged for all cases with a maximum value of residual of less than 10 9 and the parameters of all motors are found as positive. The aforementioned parameters are then used to determine the various external quantities of the motors and compare them with the corresponding actual values supplied by the manufacturer. Table II summarizes the rms errors found in T st and T max as well as in Po FL, η FL, and power factor (pf FL )at full-load speed n FL. Note that these are the information used in determining the motor parameters. Results of Table II indicate that the error in T st, P ofl, η FL, and pf FL is very insignificant (in the order of ). However, the error in T max is found as Fig. 3. Distribution of error in maximum torque as a function of hp rating , indicating that the respective equation, (A10), may not represent the actual value of maximum torque and that requires further investigation. Fig. 3 shows the distribution of error in T max against hp of all motors, and it indicates that the error is always positive. This suggests that the calculated values are higher than the corresponding actual values. The normalized torque (ratio of actual torque to full-load torque) versus slip curve of a 250-hp, 2.3-kV, 6-pole motor is shown in Fig. 4(a). The solid line represents the characteristic obtained by considering slip dependent rotor parameters, as defined by (1) and (2), while the dashed line represents the characteristic obtained by keeping the rotor parameters constant. The exploded view of the characteristics around the maximum torque point is separately shown in Fig. 4(b). It can be noticed in Fig. 4 that, in both cases, the motor has exactly the same characteristic at lower slips ( s max ) or beyond point a, as expected. Note that point a represents the maximum torque point (s max,t max ) when the rotor parameters are kept constant and at that point dt/ds = 0. However, when the rotor parameters are allowed to vary at higher slips (>s max ) according to (1) and (2), the actual maximum torque occurs at point b (or slip s max). At the transition point (when the rotor parameters start changing), the torque slip characteristic changes its course and its derivative (dt /ds) may increase suddenly. The derivative ultimately crosses the zero line at point b. The normalized torque at point a is 2.3 [see Fig. 4(b)] and is exactly the same as specified by the manufacturer. In this case, the error in T max is the normalized difference between torques at points a and b. Note that (A10) represents the torque at point a and is derived for constant rotor parameters. In order to minimize the

5 HAQUE: DETERMINATION OF NEMA DESIGN INDUCTION MOTOR PARAMETERS FROM MANUFACTURER DATA 1001 Fig. 5. Distribution of error in maximum torque against the slip ratio s r. Fig. 6. Relationship between the slip ratio s r and the reactance ratio X r. Fig. 4. Torque slip characteristics of a 250-hp motor. (a) Full characteristic. (b) Exploded view around the maximum torque point. error in T max, (A10) needs to be revised to represent the torque at point b. A procedure of revising the equation is described in the following section. V. REVISED MAXIMUM TORQUE EQUATION It is obvious in Fig. 4(b) that the error in T max depends on the deviation of s max from s max or the slip ratio s r (=s max/s max ). Fig. 5 shows the distribution of error in T max as a function of s r of all motors, and it clearly indicates that the error is almost linearly related to s r. However, the value of s r is not known in advance, and thus, it cannot be implemented during the iteration process of fsolve routine. Through vigorous investigations, it was found that the ratio of stator to rotor leakage reactance (X r = X 1 /X 20 ) has a strong correlation with s r, as can be seen in Fig. 6. By observing Fig. 6, one can propose the following relationship between s r and X r s r = s max s max = a + X r b + X r s max = [ a + Xr b + X r ] s max. (16) Fig. 7. Distribution of error in maximum torque using the revised parameters. By using the least-squares curve fitting technique, values of a and b are found as and , respectively. The solid line shown in Fig. 6 is obtained from (16) and it shows an excellent agreement with the actual values. The aforementioned values

6 1002 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 23, NO. 4, DECEMBER 2008 TABLE III RMS ERRORS IN VARIOUS EXTERNAL QUANTITIES FOR DIFFERENT VALUES OF α r AND α c TABLE IV PARAMETERS OF A 250-HP MOTOR FOR DIFFERENT VALUES OF α r AND α c of a and b are used for all the motors studied. Thus, during the iteration process of fsolve routine, s max can be evaluated through (16). By knowing s max, the revised maximum torque equation can be written as [using Fig. 2(b)] T (s max)= P ag(s max) = 3I2 2 (s max)f r (s max)r 20 ω s ω s s (17) max V 2 th (R th +f r (s max)r 20 /s max) 2 +(X th +f x (s max)x 20 ) 2. where I2 2 (s max)= Thus, the maximum torque balance equation becomes f 6 (x) =T max T (s max)=0. (18) The previous equation represents the revised maximum torque condition and it should be used to replace the original condition of (11.4). VI. FINAL RESULTS The parameters of all motors are then reevaluated after replacing (11.4) by (18) and the algorithm again converged for all cases with a maximum residual of less than The parameters of all motors are again found to be positive. With the revised parameters, the rms error in T max is reduced to ,a great improvement over the previous value of The distribution of error in T max against hp of all motors is shown in Fig. 7, and it indicates that the maximum error is less than The rms errors in other quantities (T st, Po FL, η FL, and pf FL ) are again found as very insignificant (in the order of ). Different values of α r and α c are then used in determining the parameters of all motors. Seven different cases (sets α r and α c ) are considered and the corresponding rms errors found in various quantities are summarized in Table III. Results of Table III clearly indicate that the errors are insignificant in all cases. The maximum rms error is found as and it occurs in T max for cases 3 and 7. It may be mentioned here that the use of different values of α r and α c provide different set of parameters, as shown in Table IV for a 250-hp, 2.3-kV, 2-pole motor. All sets of parameters are realistic and provide exactly the same characteristic (not shown) of the motor. A similar behavior is also observed in [9]. Note that the use of different values of α r provides different values of R 1, and hence, stator copper loss, but this is partially compensated by adjusting the value of P const, as can be seen in Table IV. Higher values of α r (and hence higher stator copper loss) provide lower values of P const, and vice versa. Similarly, in many cases, R c is omitted in the equivalent circuit and P core is lumped with P fw [2], [3]. For such a case, α c can be considered as zero. On the other hand, P fw is sometime combined with P core and represented by R c. For such a case, α c can be considered as unity. In the earlier two extreme cases (α c =0and 1), both P core and P fw are considered but at two different locations. Use of other values of α c (0 <α c < 1) simply transfers a part of the loss from one location to another, and thus, has insignificant effects on the overall characteristics of the motor. As mentioned earlier, the selection of equivalent circuit may depend on the study objective. The adjustment of rotor parameters (1) and (2) is suggested to predict the starting torque using a single-cage rotor model. In this study, it is also investigated whether the same adjustment can predict the starting current and power factor (pf). When the starting current or pf is to be predicted, it is necessary to include the respective criterion into the problem formulation. The starting current and pf criteria can be written as f 7 (x) =I 1st I 1 (1) = 0 (19) f 8 (x) =pf st cos ϕ(1) = 0. (20) Here I 1st and pf st are the manufacturer specified starting phase current and pf, respectively, and the expressions for I 1 and ϕ are given in the Appendix. The parameters of all motors are then reevaluated by replacing the starting torque criterion (11.5) by the starting current criterion (19) or the starting pf criterion (20) and the corresponding rms errors found are summarized in Table V. For comparison purpose, the results obtained by using the starting torque criterion (11.5) are also shown in the table. In all cases, the method converged with a maximum residual of less than When

7 HAQUE: DETERMINATION OF NEMA DESIGN INDUCTION MOTOR PARAMETERS FROM MANUFACTURER DATA 1003 TABLE V RMS ERRORSWHEN VARIOUSSTARTING CRITERIAARE USED the starting current criterion is used, the estimated parameters can correctly predict the starting current with an rms error of However, the rms errors in starting torque and pf are not insignificant (see Table V) because such information was not used in problem formulation. Thus, it is not fair to expect the respective correct results. The similar situation can also be observed in other two cases (when the starting torque or starting pf criterion is used). Note that the rms errors in other quantities, η FL, and pf FL ) are very insignificant (in the order of ) for all cases. When both the starting torque and starting current criteria are used by introducing one more independent variable R 1 [by removing constraint (6)], the fsolve routine end up with a large value of residual (>10) for many cases indicating that the set of nonlinear equations used in determining the parameters may not have a zero. Rest of the cases, the method converged with a small value of residual (<10 9 ), but provide unacceptable values of parameters. For example, in many cases, R 1 and P const are found as negative. Introduction of one more independent variable is possibly the main reason of providing such unrealistic results. In order to predict the starting torque, current, and pf, it is necessary to include such information in the problem formulation. For such a case, a double-cage rotor model is to be used because it has more independent parameters [9]. (P FL o VII. CONCLUSION This paper proposes a simple method of translating the manufacturer data of NEMA design A and B types of induction motors into an equivalent circuit model with slip-dependent rotor parameters. The set of nonlinear equations used to determine the parameters are derived in the form of F(x) = 0 and solved using a least-squares-based algorithm. It is found that the standard maximum torque equation may not represent the actual value of maximum torque when the rotor parameters are considered as slip dependent. The maximum torque and the corresponding slip equations are also revised in this paper to get the correct results. The proposed method is vigorously tested on more than 300 HV induction motors of various sizes. The manufacturer data that are used in evaluating the motor parameters are then determined through the equivalent circuit and are found to be in excellent agreement for all the cases studied. However, the circuit may not predict other information that was not used at all in problem formulation. The number of manufacturer data to be used depends on the number of independent variables in the equivalent circuit. Simulation results indicated that a single-cage rotor model, with slip dependent rotor parameters, can predict one of the starting characteristics (torque, current, or pf) in addition to other characteristics in the normal operating region. APPENDIX The characteristics of an induction motor can be determined from its equivalent circuit of Fig. 2(a). For a given operating slip s, the input impedance (Z in )ofthemotoris Z in (s) =Z 1 + Z sh // Z 2 (s). (A1) Here, Z 2 (s) is the effective rotor impedance and is given by Z 2 (s) = R 20(s) + jx 20 (s). (A2) s The magnitudes of stator current I 1 and rotor current I 2 are given by I 1 (s) = V 1 Z in (s) and I 2 (s) = Z sh Z sh + Z 2 (s) I 1(s). (A3) The air-gap power P ag is P ag (s) =3I2 2 (s) R 20(s). (A4) s The voltage magnitude E across the shunt branch can be written as E(s) = V 1 Z 1 I 1 (s) (cos ϕ(s) j sin ϕ(s)). (A5) The power factor angle ϕ is the same as angle of Z in. Thus ϕ(s) = Angle(Z in (s)). (A6) The output power P o at full-load slip (s FL ) can be written as P o (s FL )=(1 s FL )P ag (s FL ) P fw = 3(1 s FL )I2 2 (s FL ) R 20 (1 α c )P const. (A7) s FL Note that s FL <s max, and thus, R 2 (s FL )=R 20 according to (3). When the stray-load loss is neglected, the active power balance equation at full-load slip becomes P in (s FL )=P o (s FL )+3I1 2 (s FL )R 1 +3I2 2 (s FL )R 20 + P const. (A8) Similarly, the reactive power balance equation at full-load slip is Q in (s FL )=3I1 2 (s FL )X 1 +3I2 2 (s FL )X E2 (s FL ). X m (A9) The breakdown or maximum torque (at s = s max )is T (s max )= 3 Vth 2 2ω s R th + Rth 2 +(X th + X 20 ). (A10) 2 The starting torque (at s = 1) is T (1) = P ag(1) = 3I2 2 (1)f r (1)R 20. (A11) ω s ω s In Fig. 2(b), the magnitude of rotor current at starting can be written as V th I 2 (1) = (Rth + f r (1)R 20 ) 2 +(X th + f x (1)X 20 ). 2 (A12)

8 1004 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 23, NO. 4, DECEMBER 2008 Note that the rotor current in Fig. 2(b) is the same as in Fig. 2(a). Thus, in terms of rotor current, the stator current I 1 in Fig. 2(a) can be written as I 1 (1) = ki 2 (1) (A13) where k = 1+ f r (1)R 20 +jf x (1)X 20. Z sh Using (A13), the starting torque (A11) can be expressed as [ ] 3fr (1)R 20 T (1) = k 2 I 2 ωs 1 (1) I 1 (1) = k T (1). ω s 3f r (1)R 20 (A14) Equation (A14) indicates that the starting torque and current are not independent but related through some parameters that are determined from other information. REFERENCES [1] IEEE Task Force, Standard load models for power flow and dynamic performance simulation, IEEE Trans. Power Syst., vol.10,no.3,pp , Aug [2] IEEE Standards 112, Test Procedure for Polyphase Induction Motors and Generators. NY: IEEE, [3] P.C.Sen,Principles of Electric Machines and Power Electronics, 2nd ed. New York: Wiley, [4] G. B. Shrestha and M. H. Haque, AC Circuits and Machines. Education South Asia, Singapore: Pearson Prentice-Hall, [5] J. Pedra and F. Corcoles, Estimation of induction motor double-cage model parameters from manufacturer data, IEEE Trans. Energy Convers., vol. 19, no. 2, pp , Jun [6] J. Perdra, Estimation of typical squirrel-cage induction motor parameters for dynamic performance simulation, Inst. Elect. Eng. Proc. - Gener. Transm. Distrib., vol. 153, no. 2, pp , Mar [7] J. C. Das, Effects of momentary voltage dips on the operation of induction and synchronous motors, IEEE Trans. Ind. Appl., vol. 26, no. 4, pp , Jul./Aug [8] B. K. Johnson and J. R. Willis, Tailoring induction motor analytical models to fit known motor performance characteristics and satisfy particular study needs, IEEE Trans. Power Syst., vol. 6, no. 3, pp , Aug [9] F. Corcoles, J. Pedra, M. Salichs, and L. Sainz, Analysis of the induction machine parameter identification, IEEE Trans. Energy Convers.,vol.17, no. 2, pp , Jun [10] J. Pedra and L. Sainz, Parameter estimation of squirrel-cage induction motors without torque measurements, Inst. Elect. Eng. Proc. Electr. Power Appl., vol. 153, no. 2, pp , Mar [11] T. V. Cutsem and C. Vournas, Voltage Stability of Electric Power Systems. Norwell, MA: Kluwer, [12] J. W. Wills, G. J. Brock, and J. S. Edmonds, Derivation of induction motor models from standstill frequency response tests, IEEE Trans. Energy Convers., vol. 4, no. 4, pp , Dec [13] M. H. Haque, Estimation of three-phase induction motor parameters, Electric Power Syst. Res., vol. 26, pp , [14] J. J. Cathey, Electric Machines: Analysis and Design Applying Matlab. New York: McGraw-Hill, [15] M. Akbaba, M. Taleb, and A. Rumeli, Improved estimation of induction machine parameters, Electric Power Syst. Res., vol.34,pp.65 73,1995. [16] J. E. Brown and C. Grantham, Determination of the parameters and parameter variations of a 3-phase induction motor having a current displacement rotor, Proc. Inst. Elect. Eng., vol. 122, no. 9, pp , Sep [17] C. Grantham and D. J. McKinnon, Rapid parameter determination for induction motor analysis and control, IEEE Trans. Ind. Appl., vol. 39, no. 4, pp , Jul./Aug [18] NEMA Standards MG-1, Publication No. MG , USA. [19] Y. El-Ibiary, An accurate low-cost method for determining electric motors efficiency for the purpose of plant energy management, IEEE Trans. Ind. Appl., vol. 39, no. 4, pp , Jul./Aug [20] B. Lu, T. G. Habetler, and R. G. Harley, A survey of efficiency-estimation methods for in-service induction motors, IEEE Trans. Ind. Appl., vol.42, no. 4, pp , Jul./Aug [21] Matlab R2007a and Simulink, [22] NEMA motors and process performance cast iron motors. M. H. Haque (S 84 M 89 SM 93) was born in Dinajpur, Bangladesh. He received the B.Sc. and M.Sc. degrees in electrical engineering from Bangladesh University of Engineering and Technology, Dhaka, Bangladesh, in 1980 and 1983, respectively, and the Ph.D. degree in electrical engineering from King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, in He was earlier with the Department of Electrical and Electronic Engineering, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh, as a Lecturer for four years. During 1984, he was a Lecturer in the Department of Electrical Engineering, King Fahd University of Petroleum and Minerals (KFUPM), Dhahran, Saudi Arabia, where he became an Assistant Professor in 1989 and an Associate Professor in He was with the School of Electrical Engineering, University of South Australia, Australia, as a Senior Lecturer for three years and the Flinders University of South Australia for one year. Since 1998, he has been with the Nanyang Technological University, Singapore, where he is currently an Associate Professor in the School of Electrical and Electronic Engineering. Dr. Haque is a Fellow of the Institution of Engineers, Australia.