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squirrel-cage induction motor, torque measurement. I. INTRODUCTION. SQUIRREL-CAGE induction motors, also known as asyn- chronous motors, are used to ...
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 64, NO. 5, MAY 2015

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A Method for Measuring Torque of Squirrel-Cage Induction Motors Without Any Mechanical Sensor Wilton Lacerda Silva, Member, IEEE, Antonio Marcus Nogueira Lima, Senior Member, IEEE, and Amauri Oliveira, Member, IEEE Abstract— This paper presents a method for estimating the torque of three-phase squirrel-cage induction motors. The method is not invasive, since it does not rely on mechanical sensors and only the measurement of the stator current is required. The slip and the rotational speed of the motor are estimated from the spectrogram of the stator current signal either in stationary or in nonstationary operating conditions. The torque is estimated from the ratio of the estimated slip and the rated one. A test platform equipped with a rotational speed sensor and a dc motor for measuring the torque of a three-phase induction motor fed by a pulsewidth-modulated voltage source inverter operating under scalar and vector control was implemented to verify the correctness of the proposed method. The validity of the proposed approach is demonstrated by the close agreement between the estimated values of the speed and torque obtained with the proposed method and the ones that are obtained from direct measurement. Index Terms— Rotating machine measurements, signal processing, slot harmonics, spectral analysis, speed measurement, squirrel-cage induction motor, torque measurement.

I. I NTRODUCTION

S

QUIRREL-CAGE induction motors, also known as asynchronous motors, are used to convert electrical energy into mechanical energy. This type of motor is used in several industrial processes due to its intrinsic robustness and low maintenance requirements. To evaluate the energy conversion process, it is quite important to know the torque imposed by the electric motor to the driven mechanical system. The knowledge of this torque may reveal failures or malfunctions and thus can be exploited in monitoring systems for preventing unscheduled maintenance shutdowns or to determine the dynamic performance and the efficiency of the energy conversion process [1]. There are several techniques to measure the torque of rotating machines. One of the first proposed techniques is

Manuscript received June 16, 2014; revised October 24, 2014; accepted October 29, 2014. Date of publication December 4, 2014; date of current version April 3, 2015. This work was supported in part by the Research Support Foundation of the State of Bahia and in part by the National Council for Scientific and Technological Development through the Development Project. The Associate Editor coordinating the review process was Dr. Edoardo Fiorucci. W. L. Silva is with the Department of Education, Federal Institute of Bahia, Vitoria da Conquista 45075-265, Brazil (e-mail: [email protected]). A. M. N. Lima is with the Department of Electrical Engineering, Federal University of Campina Grande, Campina Grande 58429-140, Brazil (e-mail: [email protected]). A. Oliveira is with the Department of Electrical Engineering, Federal University of Bahia, Salvador 40110-090, Brazil (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2014.2371192

the Prony break, which can be considered as the simplest type of dynamometer. Nowadays, there are several types of dynamometers available in the market. The electric motor/generator dynamometer is usually employed to measure the torque in laboratories. The so-called torquemeter that is normally preferred in industrial applications is usually installed on the shaft that couples the motor and the load. Installation of a torquemeter is an invasive procedure, which requires the opening of the drive chain, the installation of the sensor, and the alignment of the entire mechanical tree. In some applications, specially at high-power level applications, the execution of such invasive procedure is unfeasible [2]. Some researches use other approaches, such as artificial neural network, sensorless techniques, and adaptive system model, among others, in order to evaluate torque [3]–[6]. It is also possible to use methods to evaluate the motor efficiency, as in [7] and [8] and then modify them in order to obtain motor torque. In this paper, we propose to use the slip method to estimate torque in three-phase induction motors with squirrel cages. Here, the torque is presumed to be proportional to the ratio of the measured slip s and the rated slip sr . The spectral harmonics related to the rotor slots of an induction motor that are present in stator currents will be used to accurately estimate the motor slip and rotational speed. In [9], an introductory study about the possibility of using the slip method to estimate the electromagnetic motor torque when the motor is directly fed by a pure sinusoidal ac power was presented. Now, it is investigated how to apply this approach when an induction motor is powered by a variable frequency power converter. The general equation to deal with this approach was developed. The signal-processing tool, zoom short time chirp z transform (ZSTCZT) was utilized to improve accuracy when finding the motor speed. The main advantages of this method are related to its simplicity and the possibility of being applied in systems already in operation in the field without ever interrupting its operation. It can also be used when the motor is working with a negative slip, as could be seen in motors used in the suckerrod pumps employed in artificial lift of oil [10]. Experiments using a frequency converter that can operate with scalar or vector control were conducted to verify the applicability of this approach. II. M ETHODS U SED TO E STIMATE THE T ORQUE It is possible to determine torque in three-phase induction motors, as shown in [2] and [11], considering only one degree

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of freedom of the rotor movement, using this expression Te − TL = J where J B TL Te ωm

dωm + Bωm dt

(1)

inertia of rotating masses (kg · m2 ); friction coefficient (N · m · s/rad); load torque (N · m); electromagnetic torque (N · m); mechanical angular speed of rotation (rad/s).

In [11], it is proposed to use motor parameters to find out the electromagnetic torque and thus, estimate the load torque. They presented good results in laboratory experiments, but, the drawback of this approach is related with the difficulty to reproduce it in field. Another approach applied to estimate torque is based on efficiency estimation. In this method, the torque is given by Tshaft = η

Pin ωm

(2)

where η is the efficiency and Pin is the input power. The input power can be measured using data from instantaneous currents and voltages. The speed can be measured by a tachometer or encoder and the efficiency can be found in the motor nameplate. But this efficiency is related to the motor operation at the rated power and speed. Therefore, acquiring the real motor efficiency, when, both load and speed change, can be difficult. The electromagnetic torque in three-phase induction motors can also be estimated, as shown in [12], using the following equation: √ p 3 [i a (ψc − ψb ) + i b (ψa − ψc ) + i c (ψb − ψa )] (3) Te = 6 where p is the number of pole pairs, i a , i b , and i c are motor currents and ψa , ψb , and ψc are instantaneous flux linkages. It is possible to measure the instantaneous flux linkages using search coil. The drawback here is related with the difficulty to insert the search coils into the induction motor and also with the error caused by the saturated unbalanced magnetic path. In [7], which also deals with motor efficiency, it can be seen that torque could also be estimated by the motor current data, as shown in the following expression: I ωr Tr T = Ir ω

Fig. 1. Normalized stator current as a function of the slip. Imax is the maximum value of the stator current.

(4)

where Tr is the rated torque, ωr is the rated speed, Ir is the rated current, and ω is the motor shaft speed. This method is considered easy to apply because it only requires measuring the current and speed. However, its drawback is related with the impossibility of the current becoming negative, as can be observed in Fig. 1. This curve was simulated with the model provided in [13] and [14] and shows the stator current as a function of the slip. It is possible to see that when the motor is operating with a negative slip the current will not become negative, and the estimated torque using (4) will not provide negative values as would be expected.

Fig. 2. Continuous line: typical induction motor torque-speed characteristic. Dashed line: linear torque. Nr is the rated speed, T is the motor torque, Tr is rated torque, n is the motor speed, and the synchronous speed is Ns .

In this paper, a linear relationship in the torque-speed curve is proposed to obtain the induction motor torque. Fig. 2 presents a typical torque-speed curve obtained from a model of an induction motor equivalent circuit and the region that will be considered linear is highlighted by an ellipse. The linearization is performed using two points (Ns , 0) and (Nr , Tr ), where Ns is the synchronous speed, Nr is the motor rated speed, and Tr is the rated torque. Thus, considering that the torque-speed curve can be linearized in this region, it is possible to infer the electromagnetic torque in three-phase induction motors Te as a function of rotational speed n by     Ns − n s Te = Tr = Tr . . (5) Ns − Nr sr It is possible to see that, in the considered region, torque will be proportional to the slip. Typically, the motor nameplate data does not contain the rated torque information. In the other hand, the rated power Pr and rated rotational speed are always listed. Thus, the rated torque can be given by Tr =

Pr . Nr

(6)

III. S PEED E STIMATION In almost all methodologies presented up to this point, knowing the rotational speed of the motor is very important. Here, we use the methodology of rotor slots harmonics, that are present in stator current waveform to estimate the motor speed. It can be obtained using the monitored stator line currents and performing harmonic spectral

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estimation [15]. This methodology has been widely used in various applications [16]–[20]. The rotor slot harmonic frequency can be inferred, for a healthy three-phase induction motor, by this expression   Z (1 − s) + δ f 1 f sh = (7) p where fsh rotor slot harmonic frequency, in hertz; Z number of rotor slots; s slip; p number of fundamental pole pairs; fundamental supply frequency, in hertz; f1 δ order of the stator time harmonic (δ = ±1, ±3, ±5, etc). It is possible to estimate rotational speed n in the shaft of three-phase induction motors in hertz, using the slip in the expression n = 60 f 1 (1 − s)/ p, as shown in [21] and [22], with (7) and it will be given by 60( f sh − δ f 1 ) . (8) n= Z A few techniques have been employed to infer rotational speed using the proposed methodology as also presented in [17], [18], [23], and [24]. The accuracy of the estimated rotational speed is directly related to the degree of accuracy in which it is possible to measure f 1 and f sh . Aiming to increase it, short time chirp z transform (STCZT) will be used. This technique allows us to work with signals that have its spectral contents variable in time domain. It uses chirp z transform or CZT to improve resolution without increasing very much computational complexity. It is also based on the z transform, for which the z plane can be divided into an arc of circle with angular spacing of the points as an arbitrary constant [25]. As shown in [19] and [20], this algorithm can provide a spectral analysis with better resolution in a narrow band, when compared with other methods, for example, the fast Fourier transform (FFT). STCZT can be written as X STCZT (m, k) =

N−1 

x(ν)(ν − m)A−ν W kν .

(9)

ν=0

It can be seen in (9), while x(ν) is a time function, X STCZT (m, k) is a function of associated variables with time m and frequency k. The window function (ν) is generally an even function with positive real values that has its maximum around the zero. Some examples of window functions are: 1) Hanning window; 2) Hamming window; 3) Blackman window; and 4) Gaussian window. The time-frequency resolution of STCZT is associated with the length of the window function and it is dependent of the Heisenberg–Gabor’s restriction. This means, if the window is made smaller to improve time resolution there will be a degradation in frequency resolution and vice versa. Thus, there is a compromise between conservation of the temporal localization and frequency localization. The STCZT resolution is given by f high − f low fbw = (10)  f STCZT = M −1 M −1

Fig. 3. Proposed algorithm to estimate motor slip, speed, and torque using the methodology of rotor slot harmonics.

where f low and f high are associated with the signal bandwidth that one wants to analyze. M is the number of points desired in frequency domain. Here, it can be highlighted that in contrast to the FFT, the number of points in frequency domain can be different from the number of points in time domain. Thus, as could be observed in [26], it is possible to improve frequency resolution using a zoomed-in view in the STCZT tool. This zoomed-in view means to resolve STCZT in a range of frequency ( f bw = f high − f low ) using a little M, and once the searched spectral frequency is found, we compute STCZT again, but now using a small frequency range f bw around the spectral component found. Here, to distinguish this process from others we will use the term ZSTCZT. In (7), it could be observed that both, number of rotor slots and number of fundamental pole pairs are essential to identify the rotor slot harmonic frequency. Then, once these parameters are known, the spectral frequencies f 1 and f sh can be identified. This way, expression (8) can be employed to estimate the rotational speed of the motor shaft. The final expression to evaluate this torque is obtained by substituting the expressions (6) and (8) into (5), resulting in the following expression: 

  Ns − 60 Pr Z ( f sh − δ f 1 ) . (11) Te = Nr Ns − Nr The algorithm is shown in Fig. 3 in the form of a flowchart. Here are emphasized the main steps used for the estimation of the motor’s slip, rotational speed, and torque. Initially, motor information, such as, number of fundamental pole pairs, rated speed, rated power, and number of rotor slots

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are required. Samples from the current signal are collected from one of the phases of the induction motor and are stored. Then, an observation window length and the overlap are set. The algorithm of ZSTCZT is then executed to find out the fundamental supply frequency f 1 . For this paper, it was done using the STCTZ twice. In the first try, the range of frequency went from f low = 0 Hz to fhigh = 120 Hz, assuming that the frequency produced by the converter could not go over this range. The second time, the range assumed was f bw = ±5 Hz around the frequency f 1 , previously found by the first STCZT. This was done to increase the f 1 resolution. Using the rated motor slip, it is possible to establish a search range for the f sh . Here, choosing one of the δ, it is possible to find out the rotor slot harmonic f sh and this is done using the STCZT tool, for example, assuming that a motor has 44 rotor slots, two pole pairs, rated slip 3%, fundamental supply frequency of 50 Hz, and δ = 1. Considering the motor slip goes from −1 to 3% and using (7), it is possible to determine the range to search f sh as: f shlow = [44/2(1 − 0.03) + 1]50 = 1117 Hz and f shhigh = [44/2(1 + 0.01) + 1]50 = 1161 Hz. It is not necessary to run ZSTCZT at this point, once the f sh search range is not so wide. Now, using the frequency values already found f 1 and f sh , it is possible to estimate the values of rotational speed n using the expression shown in (8). The torque is found using the expression given in (5). These results are stored, the window is slid and the search is done again. This process is executed until the last iteration is performed. The estimated speed and its uncertainty can be determined using (8) and (10), as presented nu =

60 60 ( fsh − δ f 1 ) ± ( f sh + δ f 1 ) Z Z

(12)

where  f sh is related with STCZTs resolution of the rotor slot harmonic frequency and  f 1 is related with resolution of fundamental supply frequency. In this equation, it is possible to observe that the uncertainty can be improved when choosing δ = 1. It can also be seen that it is possible to improve the resolution and uncertainty increasing the window length, but this will also increase the processing time and consequently decreasing time response. Another important factor when estimating the motor speed is related with the choice of the window length. If excessively large it could act as a low-pass filter, masking fast speed variations, as shown in [20]. This way, there must be a compromise between the window length, desired resolution, and time response. IV. E XPERIMENTAL R ESULTS Some experiments were done on the platform shown in Fig. 4(a) which, the schematic is shown in Fig. 4(b). The test bench was made up of the following. 1) A three-phase induction motor with power of 1CV, rated current Ir = 3.02 A, I p /Ir = 7.2, Nr = 1720 r/min, efficiency η = 79.5%, cos φ = 0.82, pole pairs p = 2, rotor slots Z = 44, directly fed by an ac power grid or by a frequency converter with scalar or vector control. 2) A dc motor with independent excitation controlled by a computer that allows simulation of many kinds of loads.

Fig. 4. (a) Experimental bench and (b) schematic employed to capture the current signal in one phase of the induction motor and to estimate the motor slip, speed, and torque.

3) A WEG CFW-09 variable frequency power converter that can operate with the following types of control: V/f (scalar), vector with encoder, and vector sensorless. 4) A Fluke 80i–110s ac/dc current probe that operates from dc to 100 kHz. 5) An active Chebyshev sixth-order low-pass filter with 2 kHz of cutoff frequency. 6) A DAQ NI6015 16 bits used as an analog-to-digital converter. 7) A DATAFLEX 22/20 torque measuring shaft with a nominal torque of −20 to 20 N · m and an error in linearity including hysteresis that is less than ±0.5%. It also can be used to measure the speed and can provide an output of 60 impulses/revolution. The motor can be supplied directly by an ac power grid or by a power converter. The current in one of the three phases in the induction motor is acquired by a current probe that uses a Hall effect sensor. An active low-pass filter is used as an antialiasing signal filter. This motor has its shaft coupled

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Fig. 5. (a) Spectral content of 200 ms of the current signal. Here, it is shown the harmonics of the fundamental supply frequency and the harmonics related with the rotor slots. (b) Spectrogram of a signal range. (c) Zoomed-in view of the spectrum, where the related rotor slots harmonics are highlighted.

to a dc generator through a torque and speed meter. The dc generator acts as an adjustable load to the motor. This adjustment is achieved by a system connected to a personal computer, which can simulate many kinds of loads. The torque and speed signals are conditioned before being converted into a digital form. The analog-to-digital converter, which has a 16-bits resolution, delivers the signal samples to a personal computer. An experiment was conducted to estimate motor torque. First, the current signal of a phase of the induction motor was acquired. This induction motor was powered by an ac power, with a line to line voltage of 220 V and frequency of 60 Hz and the current signal was sampled with 10 kS/s. The load variation was done by manipulation of the dc generator field current performed by a digital system, controlled by a personal computer. The spectral content shown in Fig. 5(a), where an active Butterworth second-order high-pass filter with cutoff frequency of 200 Hz was used to attenuate the f 1 frequency. It gives us a full view of the spectral behavior of this signal. The dashed lines represent the odd multiples of the fundamental supply frequency positions. The solid lines represent the positions of the rotor slot harmonics. The harmonic of the fundamental supply frequency was estimated in 60.080 ± 0.005 Hz. The temporal behavior of the spectral content is shown in Fig. 5(b). Here, it is possible to observe multiple harmonics of the fundamental supply frequency, such as the highlighted 19th and also the waveforms of the rotor

slot harmonics for δ = −3, δ = 1, and δ = 3. One can also note that there was a continuous change in frequency, over time, of the rotor slot harmonics components, while the load of the induction motor was continuously changed during the experiment. A zoomed-in view of the spectrum is shown in Fig. 5(c). Here, the highest peak is the rotor slot harmonic frequency for δ = 1. The initial and final times associated with this analysis are 0 and 200 ms, respectively. Therefore, if the following parameters are known, such as any of the rotor slot harmonics f sh , i.e., the frequency values for δ = −3, δ = 1, or δ = 3 given by the peak frequencies, the value of the supply frequency f 1 , the number of rotor slots, the number of pole pairs are known and, finally, using expression (8), it is possible to estimate rotational speed, that in the example shown in Fig. 5(c) will be 1779.10 ± 0.04 r/min. The torque can be found through expression (11) and in this case it will be 1.04 N · m. Fig. 6(a) shows the motor shaft rotational speed in revolution per minute. The solid curve represents the estimated values of the speed calculated using the methodology of rotor slot harmonics applying expression (8). An encoder with 60 impulses per revolution was used to acquire the curve represented by circles and it represents the instantaneous rotational speed. The rotational full scale speed error between these two curves is calculated using expression (13), where Nmea is the measured rotational speed signal. The result is shown graphically in Fig. 6(b). The maximum error found

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Fig. 6. Rotational speed curves. (a) Comparison between the estimated and measured rotational speed. (b) Speed error curve. The maximum error is equal to −0.026%.

Fig. 8. Torque x speed of the estimated and measured curves of the results shown in Figs. 6(a) and 7(a). Here, the synchronous speed was assumed constant and equal to Ns = 1800 r/min.

Fig. 9. Zoomed-in view of Fig. 2 in the ellipse region. Here, Nsc and Nrc are the corrected synchronous and rated speed, respectively. Fig. 7. Synchronous speed was assumed constant Ns = 1800 r/min. (a) Comparison between the estimated and measured motor torque. (b) Torque error curve. The maximum error found is 4.03%.

was −0.026%

 Nerr (%) = 100

n − Nmea Nr

 .

(13)

The electromagnetic torque can be estimated applying expression (11). Fig. 7(a) shows the result of the estimated and measured torque. The torque error was also evaluated using a similar expression (14), where the Tmea is the measured torque obtained by a torquemeter   Te − Tmea Terr (%) = 100 . (14) Tr The maximum error found was 4.03%. However, as shown in [27], the standard for the rotational speed nameplate shall not exceed 20% of the difference between the synchronous speed and the rated speed, when measured at the rated voltage, frequency, and load. Therefore, the difference between the synchronous speed and rated speed for this motor is 80 r/min, being 16 r/min, 20% of this difference. This can influence the maximum error obtained when applying this methodology. In Fig. 8, the torque-speed curve of the estimated and measured values when the synchronous speed was considered constant Ns = 1800 r/min is shown. Here, we can see a slight hysteresis in the measured torque curve. However,

it is not possible to observe this behavior in the estimated curve. As can be observed in (8), knowing the number of rotor slots is essential when making use of this methodology. However, in fact, this information is not given in the motor nameplate data. However, the number of rotor slots Z can be acquired applying, for example, an approach presented in [28]. There, the author shows an algorithm that also uses the motor current signal to find the Z . Another option, which can be used to find this parameter, is to use an optical tachometer to capture the motor speed when the system is operating in stationary mode. In parallel to this, the f1 frequency can be estimated by the ZSTCZT algorithm. As follows, (8) can be used considering n as the captured value by the tachometer and f sh can be found using the slip to define a search range, as explained in the algorithm shown in Fig. 3. When the motor is fed by a frequency converter the synchronous speed in (revolution per minute) can be determined by Nsc = 60 f 1 / p as in [13], [14], and [22]. Where the new synchronous speed Nsc is a function of the fundamental supply frequency variation f 1 . However, the other point (Nr , Tr ), necessary to establish the linear equation, will also change. In this sense, the change in Nr will be considered proportional to the change in Ns , meaning that the slope of the new linear equation will not change. The position of these new points (Nsc , 0) and (Nrc , Tr ) are shown in Fig. 9 and the

SILVA et al.: METHOD FOR MEASURING TORQUE OF SQUIRREL-CAGE INDUCTION MOTORS

Fig. 10. Torque x speed of the estimated and measured curves using the real synchronous speed.

new expression of torque can be expressed by     Nsc − n s Nsc Te = Tr = Tr . Ns − Nr Ns − Nr

(15)

As can be seen in (15), the fundamental frequency f 1 is an important parameter to estimate torque. Thus, the use of the ZSTCZT algorithm can improve accuracy in this estimation, even if this frequency is able to vary in time. To verify the applicability of this expression, the same information used to plot Fig. 8 was applied to estimate torque using now the expression (15). The result is shown in Fig. 10. Despite the slope of the measured curve being quite similar to the estimated curve, a hysteresis effect is visible. The difference in the slope is related to the motor windage and friction losses and the error related with the nameplate rated speed Nr . When an induction motor is powered by a variable frequency power converter, its fundamental supply frequency f 1 varies with the load change, so the synchronous speed Nsc will change as well. To verify if this methodology can be applied in this context, another experiment was conducted. First, a pattern of load variation was programmed at a personal computer that controls the dc generator. Then, the induction motor was powered directly from the ac power grid and the stator current data was captured. Finally, the motor was powered by a variable frequency power converter operating in scalar mode at various frequencies. The results of this experiment are shown in Fig. 11. In this figure, the maximum torque error was presented for each curve in all situations. It is possible to notice that the estimated torque presented similar behavior to that of the measured torque. Due to the use of a variable frequency power converter a change in the slope of the curve was also tracked. In this case, it is possible to conclude that the methodology can be applied. In the last experiment done, the variable frequency power converter was operated also in the vector sensorless mode with the same pattern of load variation used in the last one. The results of these experiences are shown in Fig. 12(a). There, it is possible to verify that, despite the distinct reaction of the torque-speed motor curve imposed by the operation of the variable frequency power converter in vector sensorless mode, this methodology is still able to detect this behavior. To better

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Fig. 11. Motor torque-speed curves when powered directly by an ac power grid or by a variable frequency power converter operating in scalar mode at various frequencies.

understand what happens with the behavior of the supply frequency and the synchronous speed, two graphics were shown in Fig. 12(b) and (c), respectively. One can see that there is a variation in the fundamental supply frequency f 1 in function of the load change even when the induction motor is connected directly to the ac power grid and this is reflected in a slight variation of its synchronous speed Nsc . In the other hand, when (15) is used, the variations in f 1 do not affect the results. Another important point that can be observed here, is that, the maximum error found in these experiments seem to have the same magnitude when considering (5). This can be related with the fact that we do not know which are the motor losses and rated speed error. Thus, one can realize that it is possible to estimate torque in a three-phase induction motor with a squirrel-cage rotor using this methodology with a constant or variable load and with or without variable frequency drives. Finally, it was investigated the total time spent by the algorithm applied to estimate the motor torque using this approach. The window length and signal sample rate are the most important parameters to be considered. This way, to verify how fast this algorithm can be run when it is embedded in a general purpose microcontroller, experiments were done utilizing the 32F429IDISCOVERY kit manufactured by STMicroelectronics, which has an Advanced RISC Machines Cortex-M4 core. A current signal of a motor directly powered by an ac source was captured using the internal 12-bits analogto-digital converter of the STM32F429ZIT6 microcontroller with a sample rate of 5 kS/s. A few analyses were made adopting a window length of 200 ms. The total time taken to estimate the motor torque was 180 ms. Therefore, it may be noted that this approach provides a response time greater than those shown in the study of direct torque control as in [3]–[5]. This is associated with the solution of STCZT to be strongly dependent of window length. However, it should be highlighted that when using this approach it is not necessary to know some motor parameters, such as the resistance and inductance of stator and rotor. Therefore, this can be a great advantage when using it in systems already operating in field. This way, it can be seen that it is possible to use this

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nonstationary mode. Therefore, motor torque was estimated using these speed results on the linearized torque speed curve. The general equation that can be applied even when an induction motor is powered by a variable frequency power converter was developed. The results of the estimated torquespeed curves presented great similarity to the measured curves. The drawback of this methodology is associated with the error that could exist in the rated speed and also with the fact that windage and friction losses are not known. On the other hand, when compared with others regarding the ease of use, this technique could be very helpful and it could be applied in many practical situations. R EFERENCES

Fig. 12. (a) Motor torque-speed curves when the motor was powered directly by an ac power grid or by a variable frequency power converter operating in scalar or vector sensorless mode. (b) Fundamental supply frequency. (c) Synchronous speed.

methodology as an estimator which has an acceptable response time, even when a low-performance computer system is used to run it. V. C ONCLUSION This paper has presented a new approach in the process of estimating torque in induction motors that drive loads without ever interrupting its operation. The rotor slot harmonics methodology has been applied to find the motor speed using new digital signal processing methods, which help to acquire great accuracy, even when the motor operates in

[1] 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. 1205–1210, Jul./Aug. 2003. [2] E. R. Collins, Jr., and Y. Huang, “A programmable dynamometer for testing rotating machinery using a three-phase induction machine,” IEEE Trans. Energy Convers., vol. 9, no. 3, pp. 521–527, Sep. 1994. [3] G. Terorde and R. Belmans, “Speed, flux and torque estimation of induction motor drives with adaptive system model,” in Proc. Int. Conf. Power Electron., Mach. Drives, Jun. 2002, pp. 498–503. [4] J. Faiz, M. B. B. Sharifian, A. Keyhani, and A. B. Proca, “Sensorless direct torque control of induction motors used in electric vehicle,” IEEE Trans. Energy Convers., vol. 18, no. 1, pp. 1–10, Mar. 2003. [5] K.-K. Shyu, L.-J. Shang, H.-Z. Chen, and K.-W. Jwo, “Flux compensated direct torque control of induction motor drives for low speed operation,” IEEE Trans. Power Electron., vol. 19, no. 6, pp. 1608–1613, Mar. 2004. [6] C. Bastiaensen, W. Deprez, W. Symens, and J. Driesen, “Parameter sensitivity and measurement uncertainty propagation in torque-estimation algorithms for induction machines,” IEEE Trans. Instrum. Meas., vol. 57, no. 12, pp. 2727–2732, Dec. 2008. [7] J. S. Hsu, J. D. Kueck, M. Olszewski, D. A. Casada, P. J. Otaduy, and L. M. Tolbert, “Comparison of induction motor field efficiency evaluation methods,” IEEE Trans. Ind. Appl., vol. 34, no. 1, pp. 117–125, Jan./Feb. 1998. [8] B. Lu, T. G. Habetler, and R. G. Harley, “A survey of efficiencyestimation methods for in-service induction motors,” IEEE Trans. Ind. Appl., vol. 42, no. 4, pp. 924–933, Jul./Aug. 2006. [9] W. L. Silva, A. M. N. Lima, and A. Oliveira, “Torque estimation using rotor slots harmonics on a three-phase induction motor,” in Proc. IEEE Int. Instrum. Meas. Technol. Conf. (I2MTC), Montevideu, Uruguay, May 2014, pp. 1301–1305. [10] S. G. Gibbs, “Utility of motor-speed measurements in pumping-well analysis and control,” SPE Prod. Eng., vol. 2, no. 3, pp. 199–208, 1987. [11] B. M. Wilamowski and J. D. Irwin, Power Electronics and Motor Drives, 2nd ed. Boca Raton, FL, USA: CRC Press, 2011. [12] J. S. Hsu, H. Woodson, and W. F. Weldon, “Possible errors in measurement of air-gap torque pulsations of induction motors,” IEEE Trans. Energy Convers., vol. 7, no. 1, pp. 202–208, Mar. 1992. [13] A. E. Fitzgerald, C. Kingsley, and S. D. Umans, Electric Machinery, 6th ed. New York, NY, USA: McGraw-Hill, 2003. [14] P. C. Sen, Principles of Electric Machines and Power Electronics, 2nd ed. New York, NY, USA: Wiley, 1997. [15] P. Vas, Sensorless Vector and Direct Torque Control, 1st ed. London, U.K.: Oxford Univ. Press, 1998. [16] M. Ishida and K. Iwata, “A new slip frequency detector of an induction motor utilizing rotor slot harmonics,” IEEE Trans. Ind. Appl., vol. IA-20, no. 3, pp. 575–582, May 1984. [17] A. Ferrah, K. J. Bradley, and G. M. Asher, “An FFT-based novel approach to noninvasive speed measurement in induction motor drives,” IEEE Trans. Instrum. Meas., vol. 41, no. 6, pp. 797–802, Dec. 1992. [18] S. Nandi, S. Ahmed, and H. A. Toliyat, “Detection of rotor slot and other eccentricity related harmonics in a three phase induction motor with different rotor cages,” IEEE Trans. Energy Convers., vol. 16, no. 3, pp. 253–260, Sep. 2001. [19] M. Aiello, A. Cataliotti, and S. Nuccio, “An induction motor speed measurement method based on current harmonic analysis with the chirpZ transform,” IEEE Trans. Instrum. Meas., vol. 54, no. 5, pp. 1811–1819, Oct. 2005.

SILVA et al.: METHOD FOR MEASURING TORQUE OF SQUIRREL-CAGE INDUCTION MOTORS

[20] W. L. Silva and A. Oliveira, “Analysis of spectral signatures of stator currents on a three-phase induction motor operating in non stationary mode for rotational speed and slip detection using rotor slot harmonics,” in Proc. IEEE Int. Instrum. Meas. Technol. Conf. (I2MTC), Minneapolis, MN, USA, May 2013, pp. 884–888. [21] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Machinery and Drive Systems, 2nd ed. New York, NY, USA: Wiley, 2002. [22] S. A. Nasar and I. Boldea, The Induction Machine Handbook, 1st ed. Boca Raton, FL, USA: CRC Press, 2002. [23] A. Ferrah, P. J. Hogben-Laing, K. J. Bradley, G. M. Asher, and M. S. Woolfson, “The effect of rotor design on sensorless speed estimation using rotor slot harmonics identified by adaptive digital filtering using the maximum likelihood approach,” in Proc. IEEE 32nd Ind. Appl. Conf., IAS Annu. Meeting, New Orleans, LA, USA, Oct. 1997, pp. 128–135. [24] D. Shi, P. J. Unsworth, and R. X. Gao, “Sensorless speed measurement of induction motor using Hilbert transform and interpolated fast Fourier transform,” IEEE Trans. Instrum. Meas., vol. 55, no. 1, pp. 290–299, Feb. 2006. [25] L. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp Z-transform algorithm,” IEEE Trans. Audio Electroacoust., vol. 17, no. 2, pp. 86–92, Jun. 1969. [26] I. Sarkar and A. T. Fam, “The interlaced chirp Z transform,” Signal Process., vol. 86, no. 9, pp. 2221–2232, 2006. [27] NEMA Standards Publication ANSI/NEMA MG 1 Motors and Generators, Rosslyn, VA, USA. Manuf. Assoc., 2004. [28] V. R. V. Rodeiro, A. Oliveira, and J. J. F. Cerqueira, “Detecção automática de parâmetros do motor de indução trifásico para estimação da velocidade por análise espectral,” in Proc. 16th Brazilian Autom. Conf. (CBA), Bahia, Brazil, 2006, pp. 785–790. Wilton Lacerda Silva (S’12–M’14) was born in Vitória da Conquista, Brazil, in 1971. He received the bachelor’s degree in electrical engineering from the Federal University of Minas Gerais, Belo Horizonte, Brazil, in 1994, and the master’s degree in electrical engineering from the Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil, in 1997. He is currently pursuing the Ph.D. degree with the Department of Electrical Engineering, Federal University of Bahia, Salvador, Brazil. He has been a Professor with the Department of Education, Federal Institute of Bahia, Vitória da Conquista, since 1996. His current research interests include signal processing for measuring purposes.

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Antonio Marcus Nogueira Lima (S’77–M’89– SM’05) was born in Recife, Brazil, in 1958. He received the bachelor’s and master’s degrees from the Universidade Federal da Paraíba (UFPB), Campina Grande, Brazil, in 1982 and 1985, respectively, and the Ph.D. degree from the Institut National Polytechnique de Toulouse, Toulouse, France, in 1989, all in electrical engineering. He was with Escola Técnica Redentorista, Campina Grande, from 1977 to 1982, and was a Project Engineer with Sul-America Philips, Recife, from 1982 to 1983. From 1983 to 2002, he was with the Department of Electrical Engineering, UFPB, where he became a Full Professor in 1996. He was a Coordinator of Graduate Studies at UFPB from 1991 to 1993 and 1997 to 2002. Since 2002, he has been with the Department of Electrical Engineering, Federal University of Campina Grande, Campina Grande, where he is currently a Full Professor and was the Head of the Department of Electrical Engineering, from 2002 to 2010. His current research interests include electrical machines and drive systems, power electronics, industrial automation, embedded systems, electronic instrumentation, control systems, and system identification.

Amauri Oliveira (M’88) was born in Rui Barbosa, Brazil, in 1954. He received the bachelor’s degree from the Federal University of Bahia (UFBA), Salvador, Brazil, in 1979, the master’s degree from the Federal University of Rio de Janeiro, Rio de Janeiro, Brazil, in 1982, and the Ph.D. degree from the Federal University of Paraíba, Campina Grande, Brazil, in 1997, all in electrical engineering. He is currently a Full Professor with the Department of Electrical Engineering, UFBA. His current research interests include electronic instrumentation, in particular, measurements systems, thermo-sensors, and signal processing for measuring purposes.