SENSORLESS ROTOR POSITION ESTIMATION IN A

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SENSORLESS ROTOR POSITION ESTIMATION IN A SYNCHRONOUS RELUCTANCE MOTOR P. P. Ciufo, D. Platt University of Wollongong, Department of Electrical and Computer Engineering.

Abstract The Synchronous Reluctance Motor (SRM) is an A.C. machine that has been the subject of steady research over the last several years. It is an attractive alternative to the ubiquitous squirrel cage induction motor. In order to enhance it’s appeal, methods have been investigated for the sensorless estimation of rotor position and hence remove the necessity of the shaft encoder. This paper introduces another method for rotor position estimation that adopts a statistical approach rather than a strict mathematical or model based method. The method of estimation is explained as are some early attempts to use the position data to estimate rotor velocity. 1. INTRODUCTION

position estimation down to zero speed.

The Synchronous Reluctance Motor (SRM) has received considerable attention as a suitable candidate for variable speed drive (VSD) applications [1], [2]. The overall performance of the motor appears to make it competitive with induction motors in terms of the torque available from any given frame size. However, such a drive must be operated using field oriented control, since a constant V/f drive has been found to be unstable [3].

The new estimation algorithm utilizes currents sampled at regular intervals to estimate the position of the rotor. The method works for a variety of speed ranges and is not dependent on accurate knowledge of the machine inductances, unlike many other methods. Although a high saliency ratio improves the performance of the algorithm, the technique works well for what would normally be regarded as an unacceptably low saliency ratio.

The SRM would be a less attractive device if it were not possible to determine the rotor position without a shaft mounted position sensor. As a consequence, there have been a number of publications which address the issue of sensorless rotor position estimation [4], [5], [6]. It is the aim of this paper to introduce another, novel, method of sensorless position estimation for this type of A.C. drive. The publications on this subject have, to date, attempted to model the motor or the complete drive system and develop a set of expressions that allow the direct calculation of the rotor position or velocity. The method presented in this paper is different since a statistical approach is adopted for the estimation algorithm. The SRM has features which make the sensorless position estimation process more reliable than that for the conventional squirrel cage induction motor. The SRM possesses a saliency which allows the rotor position to be sensed, since the inductances of the stator windings are dependent upon the rotor position. This feature allows rotor

For the purpose of illustration, a computer simulation has been carried out of a SRM variable speed drive. The simulation initially compared the results of the estimation algorithm with the known rotor position. In the second phase, the output of the position estimation algorithm was used in the calculation of the velocity of the drive. The defining feature of the SRM is the significant difference in the inductances measured in the direct (d-) and quadrature (q-) axes. The difference is of the order of 10:1 in machines presently being investigated. An ideal SRM has only these two inductances. A more realistic scenario would include winding resistance, rotor and stator iron losses and saturation in the d-axis. The simulation is based on the ideal model of the SRM with a saliency ratio of 10:1. This ratio is a nominal one, selected for the simulation. The value is not critical for the estimation of rotor position using the new algorithms. 2. STATISTICAL ESTIMATION BASICS

In order to understand the concept of the statistical estimation approach, first consider the path taken by the current space phasor under load at a speed in the mid-range. This is illustrated in Figure 1. This Figure shows the path taken by the current space phasor under a 0.3 p.u. load at a speed of 50 rads−1. It is clear from inspection that the high frequency behaviour of i is to move back and forth, predominately in the direction of the q-axis. This is consistent with the inductance in that axis, Lq , being many times smaller than that of the d-axis, Ld .

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Figure 2: True and Estimated Cosine of Rotor Angle 2

Clearly the estimation has a considerable amount of noise associated with it, but nevertheless it does follow the correct value. It is possible to carry out some simple filtering to improve the estimation signal. Based on the contention that the largest di will be in the direction of the q-axis, it makes sense to weight the individual estimates by the square of this distance and take the weighted average over a number of samples. By using this weighting factor, those estimates that are calculated from larger values of di contribute to the average value more than those with lesser values of di. Figure 3 shows the result of taking the ten most recent estimates.

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Figure 1: Current Space Phasor Under Load

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The voltage and current space phasors contain high and low frequency components. It is possible to filter out the low frequencies leaving a reflection of the inductances in the two axes. The filtered current space phasor consists of a series of small straight lines which, on the average, tend to be much longer in the q-axis than in the d-axis. These lines are used to form an imaginary triangle from which it is not difficult to calculate the sine and cosine of the angle that each line in the data series represents and use that as a rough estimate of the rotor position.

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Such an alogorithm is based purely on simple geometrical relationships. A little work needs to be done to get the estimate continuous and valid in the range 0 to 2π. The result of such a calculation for the cosine of the rotor position is shown in Figure 2.

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Figure 3: True and Weighted Average Cosine of Rotor Angle

All the results illustrated thus far have been taken at speeds in the middle range of the motor. At very low speeds the algorithm requires a simple modification to retain its effectiveness.

tion algorithm for a variety of saliency ratios. The statistical approach assumes that there will be larger variations in q-axis current that d-axis current as dictated by the ratio of the inductances in these two axes. In most modern machines this is certainly the case. However, an effect that reduces the effective saliency ratio of the SRM is saturation.

3. LOW SPEED PROBLEM At low speeds, the processing of the current space phasor requires special treatment. The algorithm requires information from relatively large and frequent values of di. The simple estimation algorithm does not work well at low speeds since there are many zero vector selections and these result in small values of di. To overcome this problem, the di used in the algorithm is first passed through a speed dependent thresholding filter. The task of this filter is to compare the current phasor with a speed dependent factor. If the current is larger than this factor it is passed through and forms part of the estimation algorithm. If it is not, then no new estimate is made on position.

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The filter has the characterisitic of passing only the higher values of di at the lower speeds. This has the effect of removing less useful data from the algorithm. Although the flow of data is now reduced, at lower speeds this is not a problem. The effectiveness of the thresholding technique is illustrated in Figure 4.

In the case of a simulation, the simplest way of investigating the performance of the algorithm under saturation is to test the algorithm for low saliencies. When the machine saturates in the daxis, the incremental inductance in that axis falls, resulting in a reduced saliency ratio.

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Figure 5: True and Estimated Cosine of Rotor Angle, Low Saliency Ratio

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Figure 4: True and Modified Weighted Average Cosine of Rotor Angle

This figure shows the position estimation algorithm at work for a speed reference of 5 rads−1 . The considerable amount of “noise” appearing on the non-thresholded estimation is due to the large number of zero vector selections and subsequent small changes in q-axis current. When the data is passed through the filter, the performance improves to an acceptable result. 4. LOW SALIENCY ISSUES Another important scenario, certainly worthy of investigation, is the performance of the estima-

In Figure 5 above, the simulation was performed with a saliency ratio of 3. Whilst there is a clear degredation in the quality of the estimate, a reasonable result is still produced. The estimation algorithm continues to track the rotor position. With additional signal conditioning, a more appropriate result can be obtained. This could be achieved by increasing the length of the averaging window for example. 5. VELOCITY ESTIMATION The second phase of the rotor position estimation simulation is to take the data one step further and use the position information to estimate the velocity of the drive. In many modern systems, discrete position information is made available by the use of the shaft encoder. In this instance, the position information is estimated from line current measurements. This estimated position is then used in the estimation of rotor velocity.

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There are several issues that require consideration when estimating velocity from discrete position data. These include;

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1. velocity estimation without excessive time delay, and 2. velocity estimation in the presence of noisy position information.

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Whilst the estimation algorithms provide what would appear to be a reasonable position data stream, in reality the “noise” of this signal makes direct differentation worthless. Some additional processing is required to obtain a velocity estimate with reasonable performance. Three techniques were used in this investigation. All the results presented here are for a speed reference of 50 rads−1 and a run time of 0.2 seconds (1000 samples). 5.1 Least Squares Estimation The method of least squares is well known. In its application to velocity estimation, the approach is a little different. If a LS fit is made of the position data, then the equation of the curve can be differentiated to obtain an expression for the velocity. In the LS algorithm applied to this particular work, the curve used to fit the data was a second order polynomial; x(t) = bo + b1x(t) + b2(x(t))2

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where x(t) is the arc-cosine of the rotor position. Thus after performing the curve fitting on a sample of arc-cosine estimates, an equation is obtained as a function of time, t, for the position of the rotor for that window of data. By differentiating the expression, we get; dx(t) = b1 + 2b2x(t) dt

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Figure 6: Least Squares Velocity Estimation

The result shows a definite trend to try and produce an average around the reference velocity of 50 rads−1. The estimate is still quite clearly very noisy and perhaps unsuitable for use as velocity feedback under closed loop conditions. Perhaps the saving grace is the high frequency of the noise that a drive system could not respond to. 5.2 Circular Technique A new method of estimating velocity based on position information is described here. If both the estimates of sine and cosine of rotor position are combined, a vector that forms a circle is produced. The rate at which this circle is formed is the velocity of the drive. That is; e jθ = cos θ + j sin θ

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is a unit circle. If the substitutions x = cos θ, y = sin θ and θ = ωot are made, an expression for velocity can be found by differentiation and simplification.

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Using the coefficients obtained in the first step, an estimate can now be made of the velocity of the drive. The process is repeated for consecutive, overlapping windows of data. That is, estimates are performed at every sample interval using data obtained from the previous n samples of estimated data.

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The result of using this expression for the velocity estimation is shown in the following Figure.

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The magnitude peak in Figure 8 occurs at 50 rads−1 as expected. The main criticism of this technique, although accurate results are obtained, are its excessive computational burden and the necessity for a reasonable length data set in order to track the frequency correctly. Another shortcoming would be its inability to track the velocity of the drive under dynamic conditions such as fast acceleration or shock loads.

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Figure 7: Circular Velocity Estimation

Once again, the output of the algorithm is extremely noisy. In this instance, the frequency of the noise would appear to be quite high, but the average of the signal is clearly of the order of the reference velocity. 5.3 Fourier Transform Technique A third and final method used for estimating the velocity based on discrete position data is the Fast Fourier Transform, or FFT. To use the FFT to estimate velocity from position data, the calculated sine or cosine of the rotor position is passed through the FFT algorithm. Out of the algorithm, one would expect to see a magnitude peak at a frequency corresponding to the speed of the drive. The process used is similar to that of identifying the frequency of a sinusoid in the presence of noise. 1400

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The results of all the simulations presented in this paper have been based on an ideal machine with Ld , Lq and Rs . The effects of iron loss have been ignored. The results of the velocity estimation are far from satisfactory. Although the position estimation appears to produce a reasonable output, it is not directly suitable to be used in the algorithms for velocity estimation. When these calculations are presented with ideal position data, the velocity estimation works well. Techniques to further improve the quality of the position data must continue to be investigated. It is known that a reasonable estimation of speed is required for the threshold filter. If the speed estimation is poor, then an equally poor performance of the thresholding filter would be expected. The quality of the estimated rotor position at low speeds is highly dependent on the correct selection of the current sample. Acknowledgement

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This paper has introduced a new method of rotor position estimation for the Synchronous Reluctance Motor. This method has been shown, through simulation, to be effective despite the lack of knowledge of the d- and q-axis inductances. This is an important factor in designing an algorithm to provide rotor position estimation.

The authors of this paper would like to express their acknowledgement to the Australian Research Council for providing funding for this research.

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References

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Figure 8: FFT Velocity Estimation

[1] D. A. Staton, W. L. Soong, and T. J. E. Miller, “Unified Theory of Torque Production in Switched Reluctance and Synchronous Reluctance Motors”, IEEE Transactions on Indus-

try Applications, Vol. 31, No. 2, pp. 329–336, March/April 1995. [2] J. D. Law, A. Chertok, and T. A. Lipo, “Design and Performance of Field Regulated Reluctance Machine”, IEEE Transactions on Industry Applications, Vol. 30, No. 5, pp. 1185– 1191, September/October 1994. [3] A. J. O. Cruickshank, A. F. Anderson, and R. W. Menzies, “Theory and Performance of Reluctance Motors with Axially Laminated Anisotropic Rotors”, Proceedings of the IEE, Vol. 118, No. 7, pp. 887–894, July 1971.

Shaft Sensor for Synchronous Reluctance Motor”, IEEE Transactions on Industry Applications, Vol. 30, No. 5, pp. 1202–1208, September/October 1994. [6] M. Jovanovic, R. E. Betz, and D. Platt, “Position and Speed Estimation of Sensorless Synchronous Reluctance Motor”, in Proceedings of the IEEE International Conference on Power Electronics and Drive Systems, 1995, pp. 844– 849. Addresses of the Authors

[4] T. A. Lipo T. Matsuo, “Rotor Design Optimisation of Synchronous Reluctance Motor Design”, Transactions on Energy Conversion, Vol. 9, No. 2, pp. 359–365, June 1994.

P. P. Ciufo, Department of Electrical and Computer Engineering, University of Wollongong, Northfields Ave, Wollongong, N.S.W. 2522, Australia. E-mail, [email protected].

[5] M. S. Arafeen, M. Ehsani, and T. A. Lipo, “An Analysis of the Accuracy of Indirect

D. Platt, Department of Electrical and Computer Engineering, University of Wollongong, Northfields Ave, Wollongong, N.S.W. 2522, Australia.