A sensorless speed estimation method for real time application in induction motor drives under steady-state condition is proposed. Speed harmonic component in the Space Vector Angular Fluctuation (SVAF) signal, which is caused by both dynamic and static eccentricity, is used for speed estimation. The SVAF signal is obtained from the stator currents of the induction motor. This paper is concerned with the extraction of speed information by using digital signal processing techniques. Experimental results are given to demonstrate the validity of the speed estimation method on an induction motor driven by an inverter.
Speed measurement is essential for most of the sensorless variable speed induction motor drives schemes and fault diagnoses. Advances in power semiconductor device technology have led to the widespread use of AC drives using complex control techniques such as field-oriented control or direct torque control (DTC) . The requirement of shaft mounted speed sensors renders them less attractive, because these sensors are usually expensive, and bulky, and because they increase the cost and size of the drive systems [1,2].
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Extensive work has been done on the on-line monitoring and condition assessment of induction motors. Most of the faults' diagnostic schemes presented in the literature depend upon the analysis of the motor current, flux or vibration spectrum and detection of some speed dependent harmonic components [3-6]. In many of these schemes, the accuracy of the motor speed is critical for effective condition monitoring and diagnosis [5,6].
The concept of speed estimation has been studied extensively [1-3,7-14]. Estimators, observers, and spectral analysis methods are frequently used techniques for sensorless speed estimation. Estimators, such as those in model reference adaptive system (extended Kalman filter algorithm), depend on accurate machine model and parameter estimation. However, induction motors are nonlinear and their parameters vary with time and operating conditions. Observers and spectral analysis method have a relatively long delay and data processing time that can limit real time speed measurement.
Rotor slot harmonics (RSH), which are independent of motor parameter variation and operating conditions, are extensively used to estimate motor speed . Several methods have been reported in the literature for extracting the RSH-based speed, using analog and digital signal processing (DSP) techniques [3, 9, 10,11]. However, extracting the speed information is complicated by the low level of the RSH signal relative to the fundamental harmonic and the presence of other harmonics arising both from the inverter and machine itself .
Neural networks based speed observers for speed estimation have also been discussed in [3,12]. Moreover, in  many applications of neural networks and fuzzy logic for developing speed estimators and observers for electrical motors have been presented.
It has been shown in [4, 13] that when static and dynamic eccentricity exist simultaneously (as is the case in all industrial induction motors due to the tolerance limitation), then eccentricity-related harmonics () will be present in the stator current spectrum of any three-phase induction motor irrespective of pole pairs and rotor bar number. However, in  it has been shown that the presence of RSH harmonics is primarily dependent on the number of rotor slots and the number of fundamental pole pairs of the machine, which implies that these harmonics are not present for certain combinations of rotor slots and pole pairs.
This paper proposed a new speed estimation algorithm for inverter-driven induction motors by using Space Vector Angular Fluctuation (SVAF) signal. The presented algorithm can be used to estimate motor speed in real time without the need for a speed sensor. The algorithm requires only two stator current signals and employs DSP techniques to filter and manipulate the speed-related harmonics. The algorithm is autotunable to the supply frequency. The speed associated harmonics that are used for speed estimation are the result of static and dynamic air-gap eccentricity.
2 Speed Harmonics in Stator Current
The mechanical speed information of an induction motor is embedded in the stator currents. Therefore, like condition monitoring, motor current provides a non-invasive, remote and cost-effective means for on-line speed estimation of induction motors. In addition, the filtering behaviour of the motor stator winding and the presence of current sensors in commercial drives have encouraged researchers to develop methods based on the spectral analysis of the motor current .
Induction motor stator current contains speed related harmonics, which have resulted from rotor slotting and eccentricity. They provide an accurate means of determining motor speed at virtually at any speed and load condition [10,13,14].
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In this study, eccentricity-related harmonics are used for speed estimation of induction motor driven by inverter, for two reasons. First, unlike rotor slot harmonics (RSH), eccentricity-related harmonics are always present in the stator current . Second, experiments carried out by the author have shown that eccentricity-related harmonics are not only directly present, but also the strongest ones in the nearby harmonics that appear in the SVAF signal. In contrast, RSH have lower levels than other harmonics. There are two kinds of eccentricity: static and dynamic . Static eccentricity occurs when rotor is displayed from the stator bore centre, but is still turning upon its own axis. Dynamic eccentricity occurs when the centre of the rotor is not at the centre of rotation, and therefore, the minimum airgap revolves with the rotor. In both cases the airgap is not uniform.
Harmonics in the stator current are a result of voltage harmonics induced in the winding by the airgap flux density harmonics. As a result, the frequencies of induced current harmonics can be determined by calculating the frequencies of airgap flux density harmonics when viewed from the stationary reference frame [5, 15]. The detailed derivation of airgap flux distribution and current signal are presented in , which is based on the rotating wave approach whereby the magnetic flux waves in the airgap are taken as the product of permeance and magnetomotive force (MMF) waves .
Since RSH harmonics are not used in this work, rotor and stator slotting are not considered. In an unskewed rotor the airgap permeance per unit length is given by 
where describes the variation in airgap length with angular position and time, and where both static and dynamic eccentricities are present. The permeance of an airgap bounded by smooth stator and rotor may be approximated as
where is the permeance of the average airgap length, is the degree of static eccentricity, is the degree of dynamic eccentricity and is the rotor rotational speed.
The MMF produced by the current flowing in the stator and rotor winding consists of a series of space and time harmonics. If only the stator MMF due to the nth time harmonic of current is considered, MMF will be
where p is the pole pairs and is the synchronous speed.
The flux density distribution in the airgap is given as the product of the permeance and the MMF. So, the airgap flux density is
The time component of the flux density distribution gives expressions for predicting the frequency content of the flux density waveform. These frequencies are
which can be simplified to
where s is the slip, is the fundamental supply frequency, is the nth supply harmonic and is the rotor mechanical speed in Hertz.
As these harmonics fluxes are moving relative to the stator, they should induce corresponding current harmonics in the stationary stator winding. Hence it should be possible to estimate motor speed by using the stator current.
3 Speed Frequencies in the SVAF Signal
The space vector angular fluctuation method (SVAF), which was developed from the zero cross times method (ZCT)  for induction motor failure prediction, was introduced by Kostic-Perovic et al . Like ZCT method, SVAF utilised zero crossing times of three-phase motor current waveforms as data for spectral analysis of induction motor current. Derivation of SVAF method has been presented in [6,18,19]; for this reason only the effect of speed frequencies in the SVAF spectrum will be discussed here.
Under ideal conditions the three phase stator currents of an induction motor will be
Then the space vector of balanced stator current is obtained as
where ï¡ is . In the space vector plane locus is a perfect circle. Under eccentricity, as stated in Section 2, the stator current will have additional components at frequencies of . If only the harmonic is considered to be present in the stator current, then three phase currents may be rewritten as
where is the peak value of eccentricity related harmonics. Then, the resultant space vector is
Clearly, the locus of the space vector will no longer be a perfect circle, and the rotational speed of the space vector fluctuates. The SVAF method explores these fluctuations by measuring them with respect to the balanced reference signal (Equation (8)).
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When (10) is divided by (8) we obtain
The angular fluctuation of space vector is defined as
Since and , we can apply polynomial expansion of arctan and the binomial expansion for the argument of the arctangent function in (12). Finally, the following expression can be obtained
where ï¢ is a coefficient of polynomial expansion of arctangent. This equation indicates that the real speed frequencies are directly present in the SVAF spectrum. Equation (13) also shows that the SVAF spectrum is clear from any components at the fundamental supply frequency, which makes it more useful to detect the speed related component. When compared to current, speed information obtained by the SVAF can be easily separated.
To show the ability of the SVAF method to extract speed related harmonics from the sampling current signal, simulation results are also presented here. The effects of combined static and dynamic eccentricity generate speed harmonics in the stator current. The interaction of those harmonics with the supply voltage produces eccentricity specific harmonics in the torque. This will result torque pulsation. So, the actual load torque is composed of the average load torque and eccentricity related torque oscillation . Then the total load is
where is the angular rotation of rotor. An induction motor model developed in Matlab Simulink  is used to implement the eccentricities effect. The following simulation results are obtained from this model. Here, the amplitude of is taken as 2% of full load torque. Fig. 1 shows the SVAF signal and the current spectrum obtained from the simulation of induction motor running at a constant speed. As it can be seen from the figure, in the current spectrum the supply component is the strongest one compared to the eccentricity components (). On the contrary, in the SVAF signal the speed component is the strongest one.
4 The Proposed Speed Estimation Algorithm
As it shown in the previous section, speed harmonics are not only directly present in the SVAF signal, but also the strong ones in the near by harmonics. The question here is how one can subtract this speed information from the SVAF signal. One method of getting speed is to take the FFT of the SVAF signal and then use the peak detection technique. Since this technique is time-consuming and requires long data, it is not desirable. The proposed speed estimation algorithm uses the zero-crossing method to obtained speed information from the SVAF signal. In the zero-crossing method the time between consecutive 3 zero crossings, which is a period of the main signal, is used to estimate the frequency of the signal.
For the proposed speed estimation algorithm, the time required to estimate speed is limited only by the sampling frequency of the SVAF, which depends on the supply frequency and the number of points. Three-phase system has 6 zero crossing points and the method presented in  has a sampling frequency of Hz. However, in the SVAF method the sampling frequency can be improved by using multiples of 6 points. The simulation and experimental results have shown that 12 sample points are sufficient for computation time and accuracy.
The DSP based speed estimation algorithm is shown in Fig. 2. Since the algorithm is based on DSP, it can be implemented entirely on a standard microprocessor, which is already a part of drive system. Note that if the algorithm is implemented in drive system, the required supply frequency can be directly obtained from the drive, but if the algorithm is implemented in a system, which this signal is not available, it can also be measured accurately by using the zero crossing method. Two stator currents are first sampled at 6.6 kHz. Then a digital Butterworth low-pass filter is used to filter the high frequency components produced by the drive. The filtered current signals are transferred to d-q vector form and their instantaneous vector positions are calculated. The expected angle between each consecutive SVAF sample is calculated by using the real time tracked supply frequency and the number of SVAF points . Fluctuation in the angular position are obtained by subtracting from the difference between two consecutive samples as following
is the resultant SVAF signal and contains speed- and supply-related harmonics. To eliminate unwanted harmonics, the SVAF signal is bandpass filtered whose bandwidth is sufficient for no-load to 130% load variations. Since the speed of motor varies with the supply frequency, to track this variation, the centre of the bandpass filter also has to vary with the supply frequency. The normalized corner frequencies of the designed bandpass filter depend on the sampling frequency of SVAF signal, and therefore, the ratio of corner frequencies to sampling frequency approximately remains constant. So, the designed filter automatically behaves as an adaptive filter. Afterward, the zero-crossing method is used to estimate the speed frequency from the filtered SVAF signal in real time. A disadvantage of the zero-crossing method is the requirement of a full cycle of signal for frequency estimation. For example, for a 4-pole motor supplied with 2 Hz supply frequency, the speed is approximately 1 Hz, and this means that the time requirement to estimate speed is 1s. To achieve a faster response, a half period moving window is used to estimate motor speed while fading out old data. To shorten the estimation time, every half-period of the final SVAF signal speed estimation is updated.
5 Experimental Results
In order to validate the proposed method, a voltage source inverter (VSI) fed induction motor was used. The test motor was connected to a separately excited DC generator via flexible coupling. The DC generator's output was connected to a variable resistor bank for mechanical loading. Due to the DC generator, the load torque is proportional to the motor speed. The induction motor was a 380V, 5.5 kW, 4 pole, 50Hz, 1425 rpm standard machine with a delta connected stator. An incremental shaft encoder with 1024 pulses per revolution was coupled to the motor end shaft to perform a direct measurement of the motor speed. Two hall-effect current sensors with the accuracy of ï‚±1% were used for current measurement. The currents and speed signals were sampled at 6.6 kHz simultaneously by an Eagle PCI30G data acquisition board. A visual interface program written in Agilent VEE was used to monitor the on going experiment and to save the data. The data were processed by Matlab.
Fig. 3 shows the power spectrum density (PSD) of a typical SVAF signal and its bandpass filtered version when the induction motor was running at constant speed. The supply frequency was 50Hz and motor was full-loaded. As stated above, the speed harmonic in the SVAF signal is the strongest one, which was extracted by bandpass filtering as shown in the lower panel of the figure.
Fig. 4 shows the estimated speed, measured speed by encoder and speed error when motor was running at 24.45 Hz constant speed. As illustrated in the figure, the speed of the induction motor is accurately estimated with a speed error of ï‚±0.1 Hz.
A supply frequency demand of 30 Hz to 40 Hz was applied to the tested motor when it was running at full-load. As shown in Figure 5, during the variation of supply frequency from 30 Hz to 40 Hz by drive the proposed method was able to efficiently estimate the motor speed except for a short period of time (0.2 second), where acceleration rate was very high. The error during this period of time was not higher then 0.3 Hz.
Fig. 6 shows operation of the induction motor under load variation at 40 Hz constant supply frequency. The tested load conditions were half-load and a sudden change to full-load. As it can be seen from the figure, the maximum instantaneous speed error is 0.2 Hz.
The tests were also carried out at supply frequencies as low as 1 Hz. The maximum instantaneous speed errors were in the limit of above presented results.
The estimation accuracy of presented algorithm compares favourably to the speed estimation techniques discussed in the literature. The average speed error of 5 rpm is common [3, 7-12]. For example, reported average speed error is 7 rpm and peak error is 25 rpm in , and ï‚±15 rpm in . The average errors are found to be no more than 6 rpm and the peak errors less than 18 rpm for all the cases studied.
The presented algorithm has been successfully applied to 4-pole induction motor. To check the validity of the proposed method for the 2-pole motor, several tests have been conducted. In 2-pole motor, eccentricities related harmonics are very close to the supply harmonics. Even the SVAf spectrum contained speed-related components, they are not the strongest ones and are very close to other harmonics that are present in the spectrum. For this reason, it is not possible to estimate speed for 2-pole motor by using the presented algorithm.
A sensorless speed estimation algorithm has been proposed in this paper. A test bed was used to evaluate the capabilities of the proposed algorithm for real-time speed estimation. Tests were carried out under steady-state and dynamic conditions with different load and different supply frequencies. Under all tested operating conditions the algorithm demonstrated a good performance in terms of accuracy and convergence. It has been verified at frequencies as low as 1 Hz. The proposed algorithm has the advantage that it can be implemented into drive or DSP system for real-time speed estimation, since the proposed algorithm has a low requirement on calculation time and memory space. The presented algorithm can also improve the performance and reliability of induction motor drives, because it does not require a speed sensor, extra wiring, any signal injection, detailed machine model, long data acquiring, user input of machine-specific information, and training phase.
The author would like to acknowledge the financial support provided by the Scientific Research Unit (SRU), Inonu University, Project No. 2004/14.