Torque Estimation Using Rotor Slots Harmonics on a Three-Phase Induction Motor Wilton Lacerda Silva
Antonio Marcus Nogueira Lima
Federal Institute of Bahia - IFBA Department of Electrical Engineering, Department of Electrical Engineering, o Av. Amazonas, n 3150, Zabele Federal University of Campina Grande -UFCG Federal University of Bahia - UFBA Vit´oria da Conquista - BA, Brazil Campina Grande - PB, Brazil Salvador - BA, Brazil Email: [email protected]
Email: [email protected]
Email: [email protected]
Abstract—This work presents a new approach in order to estimate torque in an induction motor which drives a load without ever interrupting its operation. Rotor slot harmonics, that are present in the stator current of an induction motor with squirrel cage rotor, are applied to estimate the torque. In particular, the speed is obtained with the rotor slot harmonics methodology and it is employed on the linearized torque-speed curve to obtain the torque. Some experiments with induction motors have been conducted. Torque and speed of the motors have been estimated. Finally, the error curves of the torque and speed have been shown. Index Terms—Non-invasive measurement techniques, rotational speed and torque estimation, rotor slot harmonics, threephase induction motor.
I. I NTRODUCTION Induction motors are intensively employed in manufacturing processes. They are robust, require low maintenance and can be used in a variety of applications. Sometimes it is important to know the torque that the motor imposes to the system that it drives. This information, for example, can be employed to determine the performance and efﬁciency of energy conversion , or also can be applied in the ﬁeld of induction motors control . However, evaluate the torque could be very laborious when the motor is already coupled to the driven equipment in the ﬁeld. It is not easy to couple a torque-measurement equipment on the system. Another fact that must be consider is the cost associated not only with the equipment, but also with its maintenance and calibration that could be required. So, this work presents a new approach in order to estimate torque in an induction motor which drives a load without ever interrupting its operation. There are two forms to measure torque: indirect and direct techniques. Indirect techniques are based on mathematical models. It is possible to estimate the motor torque by using differential equations and some motor parameters. The drawback of this methodology is related to the difﬁculty of knowing some motor parameters such as: reactance and resistance of the rotor and stator, magnetizing reactance, among others when the motor is already installed in the ﬁeld. Direct methods normally require an intervention on the system. For example, the torquemeter can be installed between the induction motor and the load. But, it is not so easy to align the torquemeter with the motor and load. Another drawback of this method is associated
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with the fact that some loads need a high start torque, so the torquemeter sensor need to be oversized, thus reduces its sensibility. An electric motor/generator dynamometer can also be employed to measure the torque. However, it is commonly used in laboratories instead of being mounted directly on production systems due to its intrinsic characteristics. Also, there are other researches that use approaches like artiﬁcial neural network, sensorless technics, adaptive system model, among others to evaluate the torque –. Finally, there are methods to evaluate the motor efﬁciency , , which can be easily modiﬁed to provide the motor torque too. This work uses the slip method that rely on the motor speed measurement. In this context, the torque is presumed to be proportional to the ratio of the measured slip s and the rated slip sr as it is shown in expression (1). s T = Tr (1) sr The main advantages of this method are related to its simplicity and the possibility of being employed when the motor is operating with negative slip, which means that the motor is operating as a generator. This work proposes to use the spectral harmonics related to the rotor slot of a induction motor that are present in stator currents to ﬁnd the motor slip, and by using (1) in the linearized region of the torque-speed motor curve, it is possible to estimate the electromechanical torque of a threephase induction motor with squirrel cage rotor, when it is operating in stationary or in non stationary mode. II. P ROPOSED
The rotational speed can be obtained by using the monitored stator line currents and performing harmonic spectral estimation . This methodology has been widely used in various applications –. The rotor slot harmonic frequency can be inferred, on a healthy induction motor, by this expression: Z (1 − s) + δ f1 (2) fsh = p where: fsh − rotor slot harmonic frequency, in Hz.
Fig. 1. The torque-speed curve of three-phase induction motor is represented by the continuous line. The linear torque is represented by the dashed line. Nr is the rated speed, Tr is rated torque and the synchronous speed is Ns .
Z − s − p − f1 − δ −
number of rotor slots. slip. number of fundamental pole-pairs. fundamental supply frequency, in Hz. order of the stator time harmonic ±1, ±3, ±5, etc.)
60 ∗ (fsh − δf1 ) Z
Some technics have been employed to infer the rotational speed by using the proposed methodology as also presented in , , , . The accuracy of the estimated rotational speed is directly related to the degree of accuracy in which it is possible to measure f1 and fsh . Aiming to increase this accuracy, the chirp z-transform (CZT) has been used. The CZT is 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 . As shown in , this algorithm can provide a spectral analysis with a better resolution in a narrow band when compared with other methods, for example the fast Fourier transform. In the expression (2), 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 and the spectral frequencies f 1 and fsh are identiﬁed, the expression (3) can be employed to estimate the rotational speed of the motor shaft. A linear relationship in torque-speed curve is proposed to obtain the induction motor torque. The Figure 1 presents the torque-speed curve obtained from an induction motor equivalent circuit. The linearization is performed by 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. So, as the torque-speed curve can be linearized in this region, it is possible to infer the electromagnetic torque in induction motor Te as a function of rotational speed n by:
Fig. 2. Schematic employed to capture the current signal in one phase of the induction motor and the motor shaft torque and rotational speed.
It is possible to estimate the rotational speed n on the induction motor shaft, in Hertz, by using the slip as shown in ,  by the following expression: n=
torque & speed meter
600 800 1000 1200 1400 1600 1800 Rotational speed (rpm)
Zoom Current Graph
4 Amplitude (A)
(N ,0) 0 0
Low pass filter
6 Phase A
Computer Digital Signal Processing
2 0 −2 −4 0
0.1 Time (s)
Fig. 3. Current signal acquired at one of the phases of the motor. The induction motor used in this experiment has been supplied by AC power of 60Hz and it has these characteristics: 2 pole pairs, 44 rotor slots, rated power of 1CV, rated speed of 1720rpm, rated current of 3.02A, efﬁciency 79.5% and cosφ=0.82.
Te = Tr
Ns − n Ns − Nr
One can verify that the expressions (1) and (4) are the same. Typically, the motor nameplate does not contain the rated torque. But it can be evaluated, for example, in (N·m), by using motor rated power Pr , in (CV), the rated rotational speed, in (rpm), by: Tr = 7024 ∗
The ﬁnal expression to evaluate this torque is obtained by substituting the expression (3) and (5) into (4). Ns − [ 60 Pr Z ∗ (fsh − δf1 )] Te = 7024 (6) Nr Ns − Nr III. R ESULTS Some experiments have been achieved on the platform whose schematic is shown in Figure 2. The motor is supplied
Rotational speed (rpm)
Window = 0.2s; Initial time = 0s; Source frequency = 59.99Hz ← 17º 1019.91Hz
8 ← 23º 1379.88Hz
1800 1780 1760
Measured with encoder. Estimated.
0 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 Frequency (Hz)
Fig. 4. Spectral content band of a piece of the current signal. The induction motor has 2 pole pairs and 44 rotor slots. The estimated rotational speed is 1751.5rpm and the estimated electromagnetic torque is 2.45N·m.
Speed error − FS (%)
0.02 Maximum error = −0.015% 0.01 0 −0.01 −0.02 0
Fig. 6. Rotational speed curves. In (a), it is shown the comparison of the estimated and measured rotational speed. In (b), it is shown the speed error curve. The maximum error is -0.015%.
1300 1400 Frequency (Hz)
Fig. 5. Spectrum band of all current signal with its temporal relationship. The motor has been directly supplied by AC power of 60Hz and the load has been varied. Here is presented some harmonics of the fundamental supply frequency (17o and 23o ) and the rotor slot harmonics frequencies for δ = −3, δ = −1, δ = 1 and δ = 3.
directly by AC power. The current in one phase of the threephase induction motor is acquired by a current probe that uses a Hall effect sensor. A low pass ﬁlter is used as an anti-aliasing signal ﬁlter. This motor has its shaft coupled to a DC generator through a torque and speed meter. The DC generator acts as an adjustable load to the motor. The torque and speed signals are conditioned before being converted into a digital form. The analog to digital converter, that has a 16bits resolution, delivers the signal samples to a personal computer. An experiment has been conducted to estimate the motor torque. Firstly, the current signal of one phase of the induction motor has been acquired. This induction motor has been supplied by AC power that has a line to line voltage of 220V and frequency of 60Hz. The motor nameplate data are: rated power of 1CV, rated speed of 1720rpm, rated current of 3.02A, efﬁciency 79.5% and cosφ=0.82. It also has 2 pole pairs and 44 rotor slots. This current signal has been sampled with 10kbps and it is shown in Figure 3(a). The load variation has been done by manipulation of the DC generator ﬁeld current. One can see that the load variation directly reﬂects on the current
signal amplitude. A zoom has been performed to exhibit the current waveform which is shown in Figure 3(b). The spectral content into a band of 1000 to 1550Hz has been performed into a part of 200ms of this current signal and it is shown in Figure 4. Here, it is possible to identify the rotor slot harmonics. The dashed lines, in blue color, represent the odd multiples of the fundamental supply frequency positions. The solid lines, in red color, represent the positions of the rotor slot harmonics. The green circles represent the sites found by the search algorithm for the frequency of the rotor slot harmonics. The highest peak is the rotor slot harmonic frequency for δ = 1. The fundamental frequency produced by the AC power supply is 59.99Hz. The initial and the ﬁnal time associated with this analysis are 0 and 200ms, respectively. So, if the values of: any of the rotor slot harmonics fsh , i.e., the frequency values for δ = −3, δ = 1 or δ = 3 given by the peaks frequencies represented by the circles in Figure 4; the value of the supply frequency f1 ; the number of rotor slots; the number of polepairs are known and, ﬁnally, by using the expression (3), it is possible to estimate the rotational speed, that in this case is 1751.5rpm. The torque can be ﬁnd through the expression (6) and its value is 2.45N·m. The temporal behavior of the spectral content is shown in Figure 5. Here, it is possible to observe the multiple harmonics of the fundamental supply frequency such as 17 o and 23o and also the waveforms of the rotor slot harmonics for δ = −3, δ = 1 and δ = 3. As it was predicted in , the harmonic associated with δ = −1 is not ideally generated. 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. The harmonics of the fundamental supply frequency did not vary. The Figure 6(a) presents the motor shaft rotational speed in rpm. The solid blue curve represents the estimated values of the speed that were calculated by the methodology of rotor slot
4 Measured with torquemeter. Estimated.
2 1 0 0
Measured with torquemeter. Estimated.
2 1 0 0
Maximum error = −10.893 0 −5
−10 −15 0
Torque error − FS (%)
Torque error − FS (%)
10 Maximum error = 5.024 5 0 −5 0
Fig. 7. Torque curves that consider Nr =1720rpm: In (a), it is shown the comparison between the estimated and the measured motor torque. In (b), it is shown the torque error curve. The maximum error found is -10.893%.
harmonics by using the expression (3). The curve represented by circles have been acquired by an encoder with 60 pulses and it represents the instantaneous rotational speed. The rotational speed error between these two curves is calculated by the expression (7), where Nmea is the measured rotational speed signal. The result is shown graphically in Figure 6(b). The maximum error found is -0.015%. n − Nmea Nerr (%) = 100 (7) Nr The electromagnetic torque can be estimated by using the expression (6). The Figure 7(a) presents the result of the estimated and measured torque. The torque error was also evaluated using a similar expression (8), where the T mea is the measured torque that was obtained by a torquemeter. Te − Tmea (8) Terr (%) = 100 Tr The maximum error found is -10.89%. However, as shown in , the standards for the rotational speed nameplate shall not exceed 20 percent of the difference between synchronous speed and rated speed when measured at rated voltage, frequency and load. So, the difference between the synchronous speed and the rated speed for this motor is 80rpm and 20% of this difference is 16rpm. So, the algorithm has been executed again considering the rated speed of 1728rpm and the torque has been estimated. The result is shown in Figure 8. Here, the maximum error is 5.02%. Thus, one can realize that it is possible to estimate the torque in a three-phase induction motor with squirrel cage rotor by using this methodology with constant or variable load. IV. C ONCLUSION This work has presented a new approach in order to estimate torque in an induction motor which drives a load without ever interrupting its operation. The rotor slot harmonics methodology has been applied to ﬁnd the motor speed. The speed error has been determined and the maximum error was just 0.015%.
Fig. 8. Torque curves that consider Nr =1728rpm: In (a), it is shown the comparison between the estimated and measured motor torque. In (b), it is shown the torque error curve. The maximum error found is 5.024%.
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