9.1.1 Cross-Correlation Scheme Derived from Optimal Detector in Additive White Gaussian Noise (AWGN) Channel

While performing motor fault detection, it is important to have a noise suppression capability where high-energy noise content dominates the low amplitude fault signatures. As an effective tool, the matched filter is often pronounced as one of the best candidates [19] in an additive white Gaussian noise (AWGN) channel. The matched filter is known as an optimal detector that maximizes the signal-to-noise ratio (SNR) in the AWGN channel. A typical filter is expressed by

where n = 1,2,…N, h_n represent the impulse response of the filter, and s_k is the input signal. The output SNR of the filter can be written as

where H = [h_N,h_N-1,…h₁], S = [s₁s₂⋯s_N],W = [w₁, w₂,⋯w_N], w_nis the sampled Gaussian noise with variance σ², and T is the vector transpose. Through the Cauchy-Schwarz inequality, the denominator in Equation (9.2) is maximized as

where c is constant. It is obvious from Equation (9.3) that the SNR of filtering is maximized when h_n = s_N−n, which is called the matched filter.

Assuming Sn is the reference signal of the inspected fault signature and x_k is the input current signal, the output of matched filter is rewritten in the form of cross-correlation as given in Equation (9.4), which is supposed to suppress noise optimally for fault signature detection. Hence, cross-cor-relation can be proposed as one of the best signal detectors for the systems distorted by Gaussian noise.

The analysis of a matched filter in continuous time can be derived in a similar manner through integration instead of summation in Equations (9.1) to (9.4) [19]. The matched filter output in continuous time can also be expressed as the cross-correlation in Equation (9.4) by replacing the summation to integration. The details of continuous-time matched filters are not covered since the implementation is based on discrete time processing.

Implementation of an algorithm on DSP is commonly limited by the memory and computing capacity of the system. The memory occupancy for cross-correlation operation is assumed negligible because it is performed in the sample sequence order of the input signal x_k in Equation (9.4), which does not need an additional signal memory buffer. The computing complexity of the cross-correlation is shown as N in Equation (9.4), which is low enough as each multiplication occurs only one time in each interrupt in normal operation of a DSP system. For the FFT-based scheme, which has been popularly used in fault diagnosis, all the signals should be inherently stored in a memory buffer for computation and the number of multiplication required is (N/2) × log(N), which is assumed not acceptable due to the overwhelming burden on DSP, especially for low cost on-line fault diagnosis systems. The inherent optimal performance in noise suppression, low memory occupation, and low computing complexity makes the cross-correlation based detection an attractive tool for on-line fault diagnosis of a motor.

Most of the specific fault signal detection schemes in literature utilize the optimal property of the matched filtering such as reference frame theory, phase sensitive detection, and any other relevant cross-correlation methods. In this chapter, the implementation of reference frame theory and phasesensitive detection as an example in an embedded DSP system is presented.

9.2.1 Reference Frame Theory for Condition Monitoring

The introduction of reference frame theory in the analysis of electrical machine systems has turned out not only to be useful in their control and analysis, but also has provided a powerful tool for condition monitoring. By judiciously choosing the reference frame, it is possible to monitor any kind of motor fault whose effects are reflected to the line current as shown in the following section. The rotating reference frame module in the software used for fault analysis can work separately and independently than the one used for motor control, which is synchronized to the fundamental harmonic vector.

9.2.2 (Fault) Harmonic Analysis of Multiphase Systems

The commonly used transformation is the polyphase to orthogonal two-phase transformation. The complex current harmonic vector describes a circular trajectory in the space vector plane as shown in Figure 9.1. Therefore, a multiphase system in phase variables transforms to a circular locus in the equivalent two-axis representation. In Figure 9.1, the radius of the circle around the origin is the peak magnitude of the inspected harmonic quantities, and the vector rotational frequency is equal to the angular frequency of phase harmonic quantities. Note that the drawings in Figure 9.1 are exag-gerated to explain the basis of the theory explicitly; indeed the magnitude of fundamental harmonic is several times higher than all line and fault har-monics. If the new rotating reference frame is defined where the axes are made to rotate at the same rate as the angular frequency of the inspected harmonic, a stationary current space vector results, where its orthogonal components are DC quantities.

If a reference frame is synchronized to a particular frequency, in the new reference frame all harmonics other than the inspected one remain as AC. The average of these AC harmonics converge to zero and have a negligible effect on the average after a sufficient time. In other words, the reference frame synchronized with fault harmonic shifts the frequency spectrum of the phase current by frequency of the fault component. The rotating frame converts only the associated fault harmonic vector to a stationary vector at zero Hertz whose projection on orthogonal base vectors are DC and the averages are nonzero in time. Thus, when the resultant fault vector modulation is normalized with respect to the fundamental vector that is computed at the synchronously rotating reference frame, the ratio gives the relative magnitude of the fault harmonic as

where i_dk and i_qk are the dq components of phase current in the rotating frame, θ₁ is the angular position of the stator reference frame, and I₁ and I_fault are the relative magnitudes of the fundamental and fault harmonic vectors, respectively. The fundamental operation of detection in Equation (9.5) constitutes multiple cross-correlation operation (matched filtering). In addition to fault harmonic magnitude calculation, the phase angle information of associated harmonic vector can also be found using the direct (d) and quadrature (q) components obtained by the proposed technique. The dq components of the harmonic vectors decouple depending on the phase angle between the rotating frame and the vector as shown in Figure 9.1. Therefore, the phase angle is formulated as

One must note that the notations, indexes, and axes of the frames might change depending on how they are defined by the user. In the literature there are different representations of reference frame theory, but the basics are the same. So reference frame theory can be further simplified for single-phase signal-based detection, which can be derived without loss of generality.

9.2.3 On-Line Fault Detection results

Experiments are done on-line using the TMS320F2812 DSP, which is employed both for inverter control and fault signature detection. Several experiments are realized under various conditions such as different rotor speeds, slip, load conditions, switching frequencies, sampling frequency, and the number of data processed.

When using DSP core for both control and fault purposes, the fault code is embedded into the main control algorithm as a subroutine that processes the instantaneously measured current data for both fundamental component and fault signature frequency. The same experiments are also repeated for line-driven motors where the DSP is responsible only for fault analysis rather than control issues. Although undersampling and oversampling are possible, generally switching frequency is accepted to be the sampling frequency of the current data to synchronize the fault subroutine with the main control. The number of data is chosen to be the same as the sampling frequency, which can be adjusted between 4K and 20K depending on the applications. The stator frequency can either be calculated or equated to the reference value depending on the control type, and the rotor speed can either be measured using encoder or estimated to update the signature frequencies in real time. Though the DSP of the inverter is used in this experiment, the very simple algorithm of reference frame theory can be implemented using a simpler microcon-troller as well.

9.3.1 Introduction

This section presents DSP-based phase-sensitive motor fault signature detection [21]. The implemented method has a powerful line current noise suppression capability while detecting the fault signatures. Because the line current of inverter-fed motors involves low order harmonics, high frequency switching disturbances, and the noise generated by harsh industrial environment, the real-time fault analyses yield erroneous or fluctuating fault signatures. This situation becomes a significant problem when SNR of the fault signature is quite low. It is theoretically and experimentally shown that the method can determine the normalized magnitude and phase information of the fault signatures even in the presence of noise, where the noise amplitude is several times higher than the signal itself.

9.3.2 Phase-Sensitive Detection

Phase-sensitive detection is based on correlation of two signals. In the correlation process, the input signal is compared with a reference signal and similarity between these signals is determined. Similarly, a lock-in detector takes a periodic reference signal and a noisy input signal, and then extracts only that part of the output signal whose frequency and phase match the reference. To see how the phase-sensitive detector works, consider a reference signal, I_ref, which is a pure sine wave with frequency of w_ref,

and the noisy fault signal,

The correlation between these two signals is given by

The generated reference signal frequency is set to be the same as the fault signal frequency; therefore some of the terms in Equation (9.9) are converted to DC as given by Equation (9.10):

If the correlation output is low pass filtered simply by averaging, only two terms survive: the DC term due to the output of the system and the noise component with frequency near the reference signal. The rest of the noise and low order harmonics disappear as shown in Equation (9.11):

The phase of the noise signal varies randomly. In order to minimize the effects of noise content at the same frequency, the phase angle difference between the reference signal and the fault signals should be minimized. There are some alternatives to maximize the low pass filtered portion of the autocorrelation function. One alternative is to track the autocorrelation function and detect the peak point where the phase angles of the reference signal and the fault signal are the same. The second and more efficient method is examining both the correlation of cosinusoidal and sinusoidal reference signals to the same phase angle instantaneously. The arctangent of the correlation ratio results in the phase angle difference between the reference signal and the fault signal. The maximum correlation degree and minimum noise effect are observed when the phase angles are equated to each other by simply adjusting the reference signal's phase angle. The similar processes are repeated for the fundamental component to calculate the correlation ratio between the fundamental and fault components to find the normalized magnitude of the fault signature.

The characteristic frequencies of the well-known motor faults are given in the literature [22–25]. The most commonly reported faults in electric machines are bearing faults, eccentricity, broken rotor bar, and stator faults. All of these faults are modeled as functions of both stator frequency and rotor speed. These two variables are mostly observed by drive systems to control the motor effectively. Therefore, the reference signals are generated according to the fault equations using the rotor speed and the supply frequency to precisely capture the associated fault signatures.

9.3.3 On-Line Experimental Results

Tests are repeated on-line using the TMS320F2812 DSP, which is employed both for inverter control and fault signature detection. When using DSP core for both control and fault purposes, the fault code is embedded into the main control algorithm as a subroutine that processes the instantaneously measured current data. The number of data is chosen to be the same as the sampling frequency, which can be adjusted between 4k and 20k depending on the applications. The stator frequency is equated to the reference value depending on the control type. The rotor speed can either be measured using the encoder or estimated to update the signature frequencies in real time. Because the embedded ADC in TMS320F2812 has 12-bit, the quantization constraints prevent sensing signals less than −65 dB. The experiments are carried out by testing broken rotor bar and eccentric motors.

The results obtained in Figure 9.5 using the DSP with 12-bit ADCs are very close to results obtained from the FFT spectrum analyzer that has a two-DSP core and a 16-bit ADC with a sampling rate of 256 kHz. The left sideband signature of an eccentric motor is measured to be −39.24 dB and −38.98 dB using the FFT analyzer and the DSP, respectively. It is reported that the fault signature magnitude is not strongly affected by the switching frequency of the inverter. Since this measurement is taken when the motor is running at the steady state, the ratio of the number of data to the switching frequency is mostly taken as unity, which provides sufficient resolution. The correlation of the fault component and the fundamental component with respect to the reference signals generated by the DSP are given in Figure 9.5c,d. It is possible to obtain smoother waveforms simply by processing more data. These on-line experimental results confirm that the method can be adapted to the real-time applications.

In order to realize on-line lock-in of the reference signal and fault signature, a few ways are possible. For instance, the phase angle difference between the reference signal and the fault signature can be calculated using the arctangent relation of the cross-correlation and the autocorrelation at each fault signature detection cycle. Next, the minimum phase angle difference point is chosen as the operating point that maximizes the correlation and minimizes the noise effects. Once this point is detected, the rest of the fault diagnosis process can be continued at this point or it can be updated at each phase difference zero crossings.

As shown in Figure 9.6a, the correlation degree of fault component is set to maximum at zero crossing of the phase difference and fixed at this point until the next zero crossing. A similar process is repeated for the fundamental component to normalize the fault component as shown in Figure 9.6b. Despite the decrease in precision, the phase angle scanning can be accelerated by increasing the reference signal phase angle increments in each drive control cycle. Since the period of phase angle scanning is in the range of minutes this method is appropriate for constant duty cycle steady-state operations.

In order to examine the motors, the duty cycles of which are continuously fluctuating, an alternative autotuning algorithm is developed. Apart from the previous method, the phase difference between the reference signal and the fault signature is continuously updated. Thanks to this method, it is possible to track fault signature not only at the steady state but during transients as well. Therefore, one can follow the dynamic characteristics of fault signatures during acceleration, deceleration, and loadings. The false error warnings can be minimized employing this method and previously determined operating point dependent on the adaptive threshold. If there are rare measurements to be made of a current magnitude that does not change significantly in time, it may be acceptable to process as many data as possible to enhance the precision of the result. However, if there are multiple measurements to be made particularly during transients, the number of processed data should be optimized. Typically, a few drive control cycles data processing time is enough at steady state and at most one or a half cycle will be sufficient during transients. It is reported that less than half a control cycle significantly degrades precision. Since the results are normalized, computation time will not affect the relative amplitude of the fault signature or the correlation degree.

In Figure 9.7, continuous tracking of the right eccentricity sideband is given. The phase lock-in is achieved in each drive control cycle by the autotuning algorithm. Using the phase-sensitive detection, the right eccentricity sideband is measured as less than |1| dB error when compared to the FFT analyzer results.

In Figure 9.8, the real-time fault signature tracks are given. In Figure 9.8a, the right eccentricity sideband variation is given from no-load to 0.33 pu load using the autotuned phase-sensitive lock-in detector. In Figure 9.8b, the broken rotor bar fault right sideband variation is given from 0.8 pu to 1.1 pu load. Because the supply frequency is already continuously available in the control algorithm and the rotor speed is measured or estimated, these parameters are used to update the fault signature frequencies in real time at various operating points. These results prove that the method has a powerful real-time fault signature tracking capability.

In this chapter, a simple noise immune real-time fault signature detection tool is presented. Since this method can easily be implemented using general-purpose microcontrollers without any additional hardware, PC, filters, and large size memory, it can be adapted to single- and multiphase drive systems.