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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Torque ripple occurs in Permanent Magnet Synchronous Motors (PMSMs) due to the non-sinusoidal flux density distribution around the air-gap and variable magnetic reluctance of the air-gap due to the stator slots distribution. These torque ripples change periodically with rotor position and are apparent as speed variations, which degrade the PMSM drive performance, particularly at low speeds, because of low inertial filtering. In this paper, a new self-tuning algorithm is developed for determining the Fourier Series Controller coefficients with the aim of reducing the torque ripple in a PMSM, thus allowing for a smoother operation. This algorithm adjusts the controller parameters based on the component's harmonic distortion in time domain of the compensation signal. Experimental evaluation is performed on a DSP-controlled PMSM evaluation platform. Test results obtained validate the effectiveness of the proposed self-tuning algorithm, with the Fourier series expansion scheme, in reducing the torque ripple.

The Permanent Magnet Synchronous Motor (PMSM) market is growing more rapidly when compared to traditional competitors because of lower cost, as well as higher efficiency and reliability. For the sake of energy savings and environmental performance, PMSMs also feature one of the highest torque to loss ratios. These motors are widely used in fast dynamic positioning systems and machine-tool components [

In order to improve the performance of PMSMs and increase its market share, the suppression of the pulsating torque has received much attention in recent years [

The second approach, which is of our interest, concentrates on using an additional control effort to compensate for the periodic torque pulsations. Some methods rely on pre-programed stator current excitation to cancel torque harmonics. However, accurate information about the PMSM parameters is required, and a small error or variation in these parameters can produce higher torque ripple due to the open-loop control. As an alternative, closed-loop control algorithms with online estimation of parameters and adaptive control algorithms have been proposed to reduce torque ripple. One possible approach relies on a closed-loop speed regulator to attenuate indirectly torque pulsations since all possible sources of torque ripple are observable from rotor speed, and hence this method has potential for complete torque ripple minimization. Repetitive Control techniques incorporate a sinusoidal control component to deal with periodic torque pulsations [

In this paper, a new self-tuning algorithm is developed for determining the Fourier Series Controller coefficients with the aim of reducing the torque ripple in a PMSM, thus allowing for a smoother operation. This algorithm adjusts the controller parameters based on the component's harmonic distortion in the time domain of the compensation signal. The estimated Fourier coefficients are used by a nonlinear controller which achieves accurate and ripple-reduced torque control. Experimental evaluation was performed on a DSP-controlled PMSM evaluation platform and test results obtained verify the effectiveness of proposed self-tuning algorithm, with the Fourier series expansion scheme, in reducing the torque ripple.

This paper is organized as follows: A model of the Permanent Magnet Synchronous Motor is presented in Section 2. The new self-tuning Fourier Coefficient Algorithm is introduced in Section 3. Section 4 describes the experimental setup, and the experimental results are presented in Section 5. Finally, in Section 6 concluding remarks are provided.

In this section, a standard PMSM model [_{abs}

In matrix form:

The stator windings voltages _{abcs}_{s}_{abcs}

Rewriting this expression in matrix form:

The stator windings are displaced by 120°, and the flux linkages Ψ_{asm}_{bsm}_{csm}_{r}_{m}

From _{s}

Defining

Hence, in Cauchy form, by using

Incorporating the transient behavior of the mechanical system, where electric torque _{e}_{L}_{m}

To find the electromagnetic torque developed _{e}_{PM}_{c}

Therefore, we have the following formula to calculate the electromagnetic torque for the three-phase

Hence:

Using the electrical angular velocity _{r}_{r}

Regarding the implicit time reparameterization to express the time functions acceleration and the speed in

To control the angular velocity, one regulates the currents fed or voltages applied to the stator windings. To maximize the electromagnetic torque developed, the motor should be fed by a balanced three-phase current set:

Generating an electromagnetic torque:

To produce the specified current, the balanced three-phase voltages are given as:

To simplify the control of PMSM, it is a common practice to transform the equations from three-phase voltages _{abcds}_{qd}_{0}_{s}

In matrix form, the mathematical model of the PMSM in the rotor reference frame is given as:

The required currents to regulate the angular velocity of PMSM and guarantee balanced operating conditions are given as:

And assuming that inductances are negligible, the applied voltages should be:

In order to understand torque ripple in PMSM we have to reconsider some assumptions from the previous model.

Flux linkages are not perfectly sinusoidal so the electromotive force differs from cosine function and applying cosine currents to the stator windings produces torque ripples. The induced non-cosine electromotive forces _{abcs}_{p}_{as}_{r}_{bs}_{r}_{cs}_{r}_{as}, e_{bs}_{cs}

Since electromagnetic torque is given by:

Using _{e}

From

Without assuming constant _{s}

The second term from

From a macroscopic viewpoint, the torque produced in a PMSM is given by [

As mentioned before, the first term appears when motor construction causes the winding inductance to vary as a function of position, and third term describes the mutual torque that is used to make the motor shaft turn. Additionally the second term describes cogging torque that appears whenever rotor magnetic flux travels through the varying reluctance of stator yokes, attempting to align with the stator teeth or poles independent of any current. When motor shaft is rotated by hand, the pulsations felt are caused by cogging torque.

As stated in the previous section, torque ripples arise from non-sinusoidal flux density distribution around the air-gap and variable magnetic reluctance due to stator slots distribution. These torque ripples change periodically with rotor position and are apparent as speed variations particularly at low speeds.

These periodic torque ripples _{r}_{r}_{0}, _{k}_{k}

Since cogging torque and harmonic components of the non-sinusoidal electromotive force depend on the slot distribution, torque ripple is a periodic function of the position and can be considered anti-symmetric and modeled by the sinusoidal components.

Considering the inertia moment of the system _{r}_{r}_{r}

To compensate for the velocity ripple a control voltage _{v}

Introducing
_{r}

For each term _{k}_{θr(t)}_{ripp}_{r}_{k}_{r}_{θr}

This algorithm permits adjusting the control voltage parameters, adapting for changes in the torque ripple, and the parameter δ allows for controlling the adjusting speed.

While the shape of the cogging torque is a complex function of motor geometry and material properties, here it is approximated by a sinusoidal function and consequently the angular position ripple is also approximated by a sinusoidal function.

Assuming that angular position is given by _{r}(t)_{0}_{0}_{0}_{0}”, and the angular position ripple “_{0}_{r}_{ripp}

For each term _{k}_{r}

In practice, _{r}

Using

So:

Recognizing that the second term is equal to zero, because can be rewritten as

Finally:

When steady state is reached, the Fourier Series Controller is applied and it provides the additional compensation so as to minimize torque ripple. Conventional PI current controllers that generate the control voltages in accordance with the field oriented control are used in the inner loop. The current controllers work with a sample time of 500 μs and gains are set as: K_{p} = 1, K_{i} = 80, all variables are considered in per unit values and the δ parameter is set to 0.02.

The performance evaluation of the controller with the proposed self-tuning algorithm is presented in the following section.

To verify the performance of the proposed self-tuning algorithm with the Fourier series expansion scheme, experiments were performed using the setup described in the preceding section. The experiments were conducted for speeds lower than 10% of the motor's nominal speed. The performance criterion used to evaluate the performance of the proposed scheme for torque ripple minimization is the variation of the angular speed determined from the angular position measurements from the encoder.

The proposed self-tuning algorithm with the Fourier series expansion scheme (for the first two terms)

In this paper has been presented an adaptive self-tuning algorithm for determining the Fourier coefficients of the controller with the aim of reducing the torque ripple in a PMSM. Its implementation is simple and represents a good alternative for minimizing torque ripple, cogging torque and non-sinusoidal electromotive torque variations due to its periodic nature. The proposed scheme does not require previous knowledge of the motor parameters. The performance of the proposed scheme has been evaluated through experimentation and test results confirm 50% speed ripple reduction (from 4.4 rpm to 2 rpm peak to peak speed ripple). Further research should be conducted to extend these results to applications where load torque varies as periodic function, which is not considered in this work.

We gratefully acknowledge the funding for the publication of this paper provided by the Mexican Council for Science and Technology (Consejo Nacional de Ciencia y Tecnología; CONACyT), under Register No. 163660. Our thanks also extend to José Enrrique Crespo- Baltar for technical advice in the mathematical development of this paper, and to Daisie Hobson for the review of the manuscript and her valuable suggestions.

Block diagram of the Fourier series controller applied to the Field Oriented Control.

Experimental setup including the TMDSHVMTRPF development and EMJ-04APB22 PMSM.

Angular position of the motor, at 273 rpm, controlled by Field Oriented Control.

Fundamental cosine term, distorted by speed ripple, for Field Oriented Control.

Speed ripple, at 273 rpm, controlled by Field Oriented Control.

Angular position of the motor, at 273 rpm, controlled by the proposed scheme, for the first two terms.

Fundamental cosine term, corrected by the proposed scheme, for the first two terms.

Speed ripple, at 273 rpm, controlled by the proposed scheme, for the first two terms.

Control signal u(θ), of the proposed scheme, for the first two terms.

Angular position of the motor, at 273 rpm, controlled by the proposed scheme, for the first four terms.

Fundamental cosine term, corrected by the proposed scheme, for the first four terms.

Speed ripple, at 273 rpm, controlled by the proposed scheme, for the first four terms.

Control signal u(θ), of the proposed scheme, for the first four terms.

Second harmonic cosine term, distorted by speed ripple, for Field Oriented Control.

Fourth harmonic consine term, distorted by speed ripple, for Field Oriented Control.

Second harmonic cosine term, corrected by the proposed schem.

Fourth harmonic consine term, corrected by the proposed scheme.

Angular position of the motor, at 80 rpm, controlled by the proposed scheme.